MH1810 Math

Chapter 2: Vectors

Imron Rosyadi

What are Vectors?

Vectors are fundamental quantities in mathematics and physics.

They possess both magnitude (size) and direction.

Tip

Think about it:

When you drive, your speed (e.g., 60 mph) is a scalar quantity.

Your velocity (60 mph North) is a vector!

Welcome to Chapter 2: Vectors! This chapter is crucial as vectors form the backbone of many concepts you’ll encounter in engineering and mathematics. Today, we’ll cover the basics of what vectors are, how to represent them, and how to perform operations like addition and scalar multiplication. We’ll also explore their representation in 2D and 3D coordinate systems, and later discuss dot products, cross products, and their applications to lines and planes.

Examples of Vectors

Vectors appear in various fields. Some common examples include:

• Acceleration: Rate of change of velocity.
• Displacement: Change in position.
• Force: A push or a pull.
• Momentum: Mass times velocity.
• Velocity: Speed in a given direction.

These examples illustrate the practical importance of vectors. For instance, when an engineer designs a bridge, they need to consider the forces acting on various parts of the structure, and these forces are vectors. Similarly, in robotics, the movement of a robot arm involves displacement and velocity vectors.

Geometrical Representation of Vectors

We can represent a vector graphically using a directed line segment.

• A directed segment from point \(A\) to point \(B\) is denoted by \(\overrightarrow{AB}\).
• The length \(|AB|\) of the segment represents the magnitude of the vector.
• The arrow indicates the direction.
• We can also use lowercase bold letters like \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\).

Vector from A to B

Geometrically, a vector is an arrow. The tail of the arrow is its initial point, and the head is its terminal point. The length of the arrow tells us the magnitude, and where it points tells us the direction. This visual representation is very intuitive for understanding vector operations.

Key Vector Concepts

Understanding these definitions is crucial:

• Equal Vectors: Two vectors are equal if they have the same magnitude and direction.
• Negative Vector: Given a vector \(\mathbf{v}\), its negative \(-\mathbf{v}\) has the same magnitude but the opposite direction.
• Zero Vector: The zero vector \(\mathbf{0}\) is a vector with zero magnitude. It has no specific direction.

The concept of equal vectors means that a vector can be translated in space without changing its identity, as long as its length and orientation remain the same. The negative of a vector is straightforward, it simply points the other way. The zero vector is important for vector algebra, acting like the number zero in scalar arithmetic.

Vector Addition (Triangle Law)

To add two vectors \(\mathbf{u}\) and \(\mathbf{v}\):

• If \(\mathbf{u} = \overrightarrow{AB}\) and \(\mathbf{v} = \overrightarrow{BC}\),
• Then \(\mathbf{u} + \mathbf{v} = \overrightarrow{AC}\).
• Simply: \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\).

This is known as the Triangle Law of Vector Addition.

Vector Addition

The Triangle Law is a visual way to add vectors. Imagine taking two steps: the first step is vector ‘u’, and the second is vector ‘v’. Your net displacement, from your starting point to your final point, is the sum ‘u + v’. This visual method helps to grasp why vector addition works this way.

Vector Addition (Parallelogram Law)

Another way to visualize vector addition:

• If \(\mathbf{u} = \overrightarrow{OA}\) and \(\mathbf{v} = \overrightarrow{OB}\),
• Then \(\mathbf{u} + \mathbf{v}\) is the diagonal of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\) starting from \(O\).

This is the Parallelogram Law of Vector Addition.

Parallelogram Law

The Parallelogram Law is equivalent to the Triangle Law. If you imagine shifting vector ‘v’ so its tail meets the head of ‘u’, you form the triangle. The parallelogram law is particularly useful when vectors originate from the same point. It highlights the commutative property of vector addition.

Vector Subtraction

Vector subtraction can be understood as adding the negative of a vector:

• \(\mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v})\).
• If \(\mathbf{u} = \overrightarrow{OA}\) and \(\mathbf{v} = \overrightarrow{OB}\),
• Then \(\overrightarrow{AB} = \mathbf{v} - \mathbf{u}\).
• Simply: \(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}\).

Vector Subtraction

Vector subtraction is essentially a special case of addition. By understanding that subtracting ‘v’ is the same as adding ‘-v’, we can apply the same geometric principles. The image clearly shows that the vector from the tip of ‘u’ to the tip of ‘v’ (when both start from the origin) represents ‘v - u’. This is very useful for finding displacement between two points in space.

Scalar Multiplication

When a vector \(\mathbf{v}\) is multiplied by a scalar \(\lambda\) (a real number):

• If \(\lambda > 0\), \(\lambda \mathbf{v}\) is a vector in the same direction as \(\mathbf{v}\) but with magnitude \(\lambda\) times that of \(\mathbf{v}\).
• If \(\lambda < 0\), \(\lambda \mathbf{v}\) is a vector in the opposite direction of \(\mathbf{v}\) with magnitude \(|\lambda|\) times that of \(\mathbf{v}\).
• If \(\lambda = 0\), \(0 \mathbf{v} = \mathbf{0}\).

Note

Any vector in the same direction as \(\mathbf{v}\) can be written as \(\lambda \mathbf{v}\) for some \(\lambda > 0\).

Scalar multiplication scales the vector. If you multiply a velocity vector by 2, the speed doubles, but the direction remains the same. If you multiply by -1, the speed stays the same, but the direction reverses. This operation is fundamental for concepts like parallel vectors and scaling forces.

Properties of Vector Operations

Vector addition and scalar multiplication follow several important properties:

• Commutativity of Addition: \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\)
• Associativity of Addition: \(\mathbf{u} + (\mathbf{v} + \mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w}\)
• Associativity of Scalar Multiplication: \(\lambda (\mu \mathbf{v}) = (\lambda \mu)\mathbf{v}\), where \(\lambda, \mu \in \mathbb{R}\)
• Distributivity (Scalar over Vector Addition): \((\lambda + \mu) \mathbf{v} = \lambda \mathbf{v} + \mu \mathbf{v}\)
• Distributivity (Scalar Multiplication over Vector Addition): \(\lambda (\mathbf{u} + \mathbf{v}) = \lambda \mathbf{u} + \lambda \mathbf{v}\)

These properties are analogous to those of real numbers and are essential for manipulating vector equations. They ensure that vector algebra behaves in a consistent and predictable manner, allowing us to simplify expressions and solve problems.

Vectors in 2-Space (\(\mathbb{R}^2\))

In a Cartesian coordinate system, we can represent a vector \(\mathbf{v}\) in 2-space:

• Initial point at the origin \(O=(0,0)\).
• Terminal point \((v_1, v_2)\).
• Written as \(\mathbf{v} = (v_1, v_2)\) or \(\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j}\).
  • \(\mathbf{i}\) is the unit vector along the positive x-axis: \((1,0)\).
  • \(\mathbf{j}\) is the unit vector along the positive y-axis: \((0,1)\).
• Column vector form: \(\mathbf{v} = \left( \begin{array}{l} v_{1} \\ v_{2} \end{array} \right)\).

Vectors in 2-Space

Representing vectors in a coordinate system allows us to perform calculations algebraically. The components \(v_1\) and \(v_2\) tell us how much the vector extends along the x-axis and y-axis respectively. The unit vectors ‘i’ and ‘j’ provide a basis for expressing any 2D vector.

Vector Operations in \(\mathbb{R}^2\)

Let \(\mathbf{u} = (u_1, u_2)\) and \(\mathbf{v} = (v_1, v_2)\).

1. Zero Vector: \(\mathbf{u} = (0,0)\) if and only if \(u_1 = 0\) and \(u_2 = 0\).
2. Equality: \(\mathbf{u} = \mathbf{v}\) if and only if \(u_1 = v_1\) and \(u_2 = v_2\).
3. Addition: \(\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2)\).
4. Scalar Multiplication: \(\lambda \mathbf{u} = (\lambda u_1, \lambda u_2)\), where \(\lambda \in \mathbb{R}\).

These rules make vector arithmetic straightforward in component form. We essentially treat each component independently for addition and scalar multiplication, which is a powerful simplification.

Vectors in 3-Space (\(\mathbb{R}^3\))

In 3-space, we use a rectangular coordinate system with three mutually perpendicular axes.

• Right-handed system: Imagine your right-hand palm along the positive x-axis, rotate \(90^{\circ}\) towards the positive y-axis, your thumb points in the positive z-axis.
• A vector from \(P_1=(x_1,y_1,z_1)\) to \(P_2=(x_2,y_2,z_2)\) is \(\overrightarrow{P_1P_2} = (x_2-x_1, y_2-y_1, z_2-z_1)\).
• If starting from origin \(O=(0,0,0)\) to \(P=(x_0,y_0,z_0)\), then \(\overrightarrow{OP} = (x_0, y_0, z_0)\).
• Components: \(x_0, y_0, z_0\).
• Also written as \(\mathbf{v} = x_0\mathbf{i} + y_0\mathbf{j} + z_0\mathbf{k}\).
  • \(\mathbf{i}=(1,0,0)\), \(\mathbf{j}=(0,1,0)\), \(\mathbf{k}=(0,0,1)\) are unit vectors along x, y, z axes.
• Column vector form: \(\mathbf{v} = \left( \begin{array}{l}x_0\\ y_0\\ z_0 \end{array} \right)\).

Vectors in 3-Space

The transition from 2D to 3D is a natural extension. The right-handed system is a convention that ensures consistent results in operations like the cross product, which we’ll discuss later. The components and basis vectors extend naturally to the third dimension.

Vector Operations in \(\mathbb{R}^3\)

Let \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\).

1. Zero Vector: \(\mathbf{u} = (0,0,0)\) if and only if \(u_1 = u_2 = u_3 = 0\).
2. Equality: \(\mathbf{u} = \mathbf{v}\) if and only if \(u_1 = v_1\), \(u_2 = v_2\), and \(u_3 = v_3\).
3. Addition: \(\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3)\).
4. Scalar Multiplication: \(k\mathbf{u} = (ku_1, ku_2, ku_3)\), where \(k \in \mathbb{R}\).

Just like in 2D, operations in 3D are performed component-wise. This simplicity is one of the advantages of using coordinate systems.

Length (Norm or Magnitude) of a Vector

The length of a vector is calculated using the Pythagorean theorem.

• In 2-space: For \(\mathbf{u} = (u_1, u_2)\), the length is \[\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2}\]
• In 3-space: For \(\mathbf{u} = (u_1, u_2, u_3)\), the length is \[\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2}\]

Note

The distance \(d\) between two points \(P=(u_1,u_2,u_3)\) and \(Q=(v_1,v_2,v_3)\) is the norm of the vector \(\overrightarrow{PQ}\): \(d = \sqrt{(v_1-u_1)^2 + (v_2-u_2)^2 + (v_3-u_3)^2}\).

The length or norm of a vector represents its magnitude, irrespective of its direction. It’s always a non-negative scalar. The formula is a direct application of the Pythagorean theorem, extended to three dimensions. It’s also important to note that \(\|k\mathbf{u}\| = |k|\|\mathbf{u}\|\), meaning scalar multiplication scales the magnitude by the absolute value of the scalar.

Unit Vectors

A unit vector is a vector of length 1.

• Example: \(\mathbf{u} = \left(-\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\right)\) is a unit vector because \(\|\mathbf{u}\| = \sqrt{\left(-\frac{1}{\sqrt{2}}\right)^2 + 0^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + 0 + \frac{1}{2}} = \sqrt{1} = 1\).
• The standard unit vectors are: \(\mathbf{i} = (1, 0, 0)\), \(\mathbf{j} = (0, 1, 0)\), and \(\mathbf{k} = (0, 0, 1)\).
• For any nonzero vector \(\mathbf{u}\), the unit vector in the direction of \(\mathbf{u}\) is \[\hat{\mathbf{u}} = \frac{1}{\|\mathbf{u}\|} \mathbf{u}\]

Unit vectors are incredibly useful because they purely represent direction. Any vector can be expressed as its magnitude multiplied by its unit vector. This separation of magnitude and direction simplifies many calculations and concepts. Normalizing a vector (turning it into a unit vector) is a common operation.

Dot Product (Scalar Product)

The dot product of two non-zero vectors \(\mathbf{u}\) and \(\mathbf{v}\) is defined as:

\[\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta\] where \(\theta \in [0, \pi]\) is the angle between the two vectors.

If either \(\mathbf{u}\) or \(\mathbf{v}\) is a zero vector, then \(\mathbf{u} \cdot \mathbf{v} = 0\).

Angle between vectors

The dot product is a scalar quantity that tells us something about the “alignment” of two vectors. If they point in the same general direction, the dot product is positive. If they are perpendicular, it’s zero. If they point in opposite general directions, it’s negative. This makes it extremely useful for finding angles and checking for orthogonality.

Properties of the Dot Product

1. Scalar Result: The dot product of two vectors is a real number (scalar).
2. Commutativity: \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\).
3. Self-Dot Product: For any vector \(\mathbf{u}\), \(\mathbf{u} \cdot \mathbf{u} = \|\mathbf{u}\|^2\).
4. Orthogonality: If \(\mathbf{u}\) and \(\mathbf{v}\) are non-zero vectors, then \(\mathbf{u} \cdot \mathbf{v} = 0 \Leftrightarrow \mathbf{u} \perp \mathbf{v}\) (they are perpendicular).
5. Scalar Multiplier: \((\lambda \mathbf{u}) \cdot (\mu \mathbf{v}) = (\lambda \mu)(\mathbf{u} \cdot \mathbf{v})\).
6. Distributivity over Addition: \(\mathbf{u} \cdot (\mathbf{v} + \mathbf{c}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{c}\).
7. Distributivity over Addition (II): \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{c} = \mathbf{u} \cdot \mathbf{c} + \mathbf{v} \cdot \mathbf{c}\).

These properties are vital for manipulating and simplifying expressions involving dot products. Property 4, in particular, is a powerful tool for determining if two vectors are perpendicular without knowing the angle explicitly.

Dot Product in Component Form

For vectors in coordinate systems, the dot product is calculated as the sum of the products of their corresponding components.

• In 2-space: If \(\mathbf{u} = \left( \begin{array}{l} u_{1} \\ u_{2} \end{array} \right)\) and \(\mathbf{v} = \left( \begin{array}{l} v_{1} \\ v_{2} \end{array} \right)\), then \[\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2\]
• In 3-space: If \(\mathbf{u} = \left( \begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \end{array} \right)\) and \(\mathbf{v} = \left( \begin{array}{l} v_{1} \\ v_{2} \\ v_{3} \end{array} \right)\), then \[\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3\]

This algebraic definition of the dot product is incredibly practical. It allows us to calculate the dot product quickly without needing to know the angle or the magnitudes explicitly beforehand. This formula can be derived using the properties of the dot product and the definitions of unit vectors.

Angle Between Two Vectors

The dot product formula provides a direct way to find the angle \(\theta\) between two non-zero vectors \(\mathbf{u}\) and \(\mathbf{v}\):

\[\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}\]

Therefore, \[\theta = \cos^{-1}\left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}\right)\]

This formula is one of the most important applications of the dot product. It allows us to quantify the angular relationship between any two vectors, which is essential in physics for problems involving force and motion, and in computer graphics for lighting calculations.

Application: Work Done

One key application of the dot product is calculating work done by a constant force.

• Force (\(\mathbf{F}\)) and displacement (\(\mathbf{D}\)) are vector quantities.
• The work done (\(W\)) by a constant force \(\mathbf{F}\) acting through a displacement \(\mathbf{D}\) is the dot product: \[W = \mathbf{F} \cdot \mathbf{D}\]

Work Done

In physics, work is done only when a force causes displacement in the direction of the force. If the force is perpendicular to the displacement, no work is done. The dot product naturally captures this, as the cosine term in its definition accounts for the alignment of the force and displacement vectors.

Projection of a Vector

The projection of vector \(\mathbf{u}\) onto vector \(\mathbf{v}\) gives the component of \(\mathbf{u}\) that lies in the direction of \(\mathbf{v}\).

• Length of projection of \(\mathbf{u}\) onto \(\mathbf{v}\) (scalar projection): \[\left|\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|}\right|\]
• Vector projection of \(\mathbf{u}\) onto \(\mathbf{v}\), denoted \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\): \[\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|}\right) \hat{\mathbf{v}} = \left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2}\right) \mathbf{v}\] where \(\hat{\mathbf{v}}\) is the unit vector in the direction of \(\mathbf{v}\).

Vector Projection

Vector projection is a powerful tool in engineering, for example, to decompose a force vector into components parallel and perpendicular to a surface, or to determine how much of a signal aligns with another signal. The scalar projection tells you “how much” of ‘u’ is in the direction of ‘v’, while the vector projection gives you the actual vector component.

Cross Product (Vector Product)

The cross product (or vector product) of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is defined as:

\[\mathbf{u} \times \mathbf{v} = (\|\mathbf{u}\| \|\mathbf{v}\| \sin \theta) \hat{\mathbf{n}}\] where:

• \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\) (\(0 \le \theta \le \pi\)).
• \(\hat{\mathbf{n}}\) is a unit vector perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\), determined by the right-hand rule.

Caution

The cross product is only defined for vectors in 3-space (\(\mathbb{R}^3\)).

Right-Hand Rule

Unlike the dot product, the cross product results in a vector. Its direction is given by the right-hand rule: if you curl the fingers of your right hand from ‘u’ to ‘v’, your thumb points in the direction of ‘u x v’. The magnitude of the cross product is related to the area of the parallelogram formed by ‘u’ and ‘v’.

Properties of the Cross Product

1. Vector Result: The cross product of two vectors is a vector.
2. Parallel Vectors: If \(\mathbf{u}\) and \(\mathbf{v}\) are parallel, then \(\theta = 0\) or \(\theta = \pi\), so \(\mathbf{u} \times \mathbf{v} = \mathbf{0}\).
3. Anti-Commutativity: \(\mathbf{u} \times \mathbf{v} = -\mathbf{v} \times \mathbf{u}\).
4. Distributivity over Addition: \(\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}\).
5. Scalar Multiplier: \((\lambda \mathbf{u}) \times (\mu \mathbf{v}) = (\lambda \mu)(\mathbf{u} \times \mathbf{v})\).

Note

Special Cross Products of Unit Vectors:

\(\mathbf{i} \times \mathbf{i} = \mathbf{j} \times \mathbf{j} = \mathbf{k} \times \mathbf{k} = \mathbf{0}\)

\(\mathbf{i} \times \mathbf{j} = \mathbf{k}\); \(\mathbf{j} \times \mathbf{k} = \mathbf{i}\); \(\mathbf{k} \times \mathbf{i} = \mathbf{j}\)

(Follow a cyclic order for positive results, reverse for negative)

The anti-commutative property (Property 3) is a key distinction from scalar multiplication and dot product. The properties involving unit vectors are critical for deriving the component form of the cross product.

Cross Product in Component Form (Determinant Formula)

If \(\mathbf{u} = \left( \begin{array}{l} u_{1} \\ u_{2} \\ u_{3} \end{array} \right)\) and \(\mathbf{v} = \left( \begin{array}{l} v_{1} \\ v_{2} \\ v_{3} \end{array} \right)\), then the cross product can be computed using a determinant:

\[\mathbf{u} \times \mathbf{v} = \left| \begin{array}{c c c} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{array} \right|\]

Expanding the determinant gives: \[\mathbf{u} \times \mathbf{v} = (u_2 v_3 - u_3 v_2) \mathbf{i} - (u_1 v_3 - u_3 v_1) \mathbf{j} + (u_1 v_2 - u_2 v_1) \mathbf{k}\]

This determinant formula provides a systematic way to calculate the cross product algebraically. It’s especially useful when dealing with complex vectors. Remember the alternating signs for the \(\mathbf{j}\) component in the expansion.

Applications of the Cross Product

1. Perpendicular Vector: \(\mathbf{u} \times \mathbf{v}\) is a vector perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\). This is crucial for finding normal vectors to planes.

2. Area of Parallelogram: The magnitude of the cross product gives the area of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\): \[\text{Area} = \|\mathbf{u} \times \mathbf{v}\|\]

3. Area of Triangle: The area of a triangle with sides \(\mathbf{u}\) and \(\mathbf{v}\) is half the area of the parallelogram: \[\text{Area} = \frac{1}{2} \|\mathbf{u} \times \mathbf{v}\|\]

These applications highlight the geometric significance of the cross product. Finding a perpendicular vector is fundamental for defining planes, and calculating areas of geometric shapes directly from vector components is a powerful tool in geometry and physics.

Scalar Triple Product

For three vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) in \(\mathbb{R}^3\): \[\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\] This is called the scalar triple product.

It can be computed as a determinant: \[\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \left| \begin{array}{c c c} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{array} \right|\]

Note

Cyclic property: \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \mathbf{v} \cdot (\mathbf{w} \times \mathbf{u}) = \mathbf{w} \cdot (\mathbf{u} \times \mathbf{v})\).

The scalar triple product combines both the dot and cross product. It results in a scalar value. The order of operations is crucial: first compute the cross product, then the dot product.

Applications of the Scalar Triple Product

1. Volume of Parallelepiped: The absolute value of the scalar triple product, \(|\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})|\), represents the volume of the parallelepiped formed by the three vectors.

2. Coplanarity: If the scalar triple product \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = 0\), then:

   • At least one of the vectors is a zero vector, OR
   • The three vectors are coplanar (they lie on the same plane).

The scalar triple product has powerful geometric interpretations. Its value gives the volume of a 3D solid, and its zero value is a definitive test for whether three vectors lie in the same plane, a concept critical in geometry and mechanics.

Lines in 3-Space: Vector Equation

A line \(\ell\) in space is uniquely determined by:

• A point on the line with position vector \(\mathbf{r}_0\).
• A direction vector \(\mathbf{v}\) parallel to the line.

The vector equation of the line \(\ell\) is: \[\ell: \mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{v}, \quad t \in \mathbb{R}\] Here, \(\mathbf{r}(t)\) is the position vector of any point on the line, and \(t\) is a scalar parameter.

Line in 3D

This vector equation is a parametric representation of the line. As ‘t’ varies, ‘r(t)’ traces out all points on the line. It’s very intuitive: you start at a point ‘r0’ and move in the direction of ‘v’ for a distance scaled by ‘t’.

Lines in 3-Space: Parametric and Cartesian Equations

Given \(\mathbf{r}_0 = \left( \begin{array}{l}x_0\\ y_0\\ z_0 \end{array} \right)\) and \(\mathbf{v} = \left( \begin{array}{l}v_{1}\\ v_{2}\\ v_{3} \end{array} \right)\), any point \((x,y,z)\) on the line has a position vector \(\mathbf{r} = \left( \begin{array}{l} x \\ y \\ z \end{array} \right)\).

• Parametric Equations: \[x = x_0 + t v_1\] \[y = y_0 + t v_2\] \[z = z_0 + t v_3\] for \(t \in \mathbb{R}\).

• Cartesian (Symmetric) Equations: If \(v_1, v_2, v_3 \neq 0\), we can solve for \(t\): \[\frac{x - x_0}{v_1} = \frac{y - y_0}{v_2} = \frac{z - z_0}{v_3}\]

Warning

If any \(v_i = 0\), that component’s equation becomes constant (e.g., \(x=x_0\) if \(v_1=0\)), and it cannot be part of the symmetric form.

The parametric equations explicitly show each coordinate’s dependence on the parameter ‘t’. The Cartesian form is useful because it removes the parameter, providing a direct relationship between x, y, and z. The warning about \(v_i=0\) is important for correctly interpreting the Cartesian form; for example, if \(v_1=0\), the line lies in a plane \(x=x_0\).

Angle Between Two Lines

To find the angle between two lines \(\ell_1\) and \(\ell_2\):

• Let \(\mathbf{v}_1\) and \(\mathbf{v}_2\) be their respective direction vectors.
• The angle \(\theta\) between the lines is the acute angle between their direction vectors. \[\theta = \cos^{-1} \left| \frac{\mathbf{v}_1 \cdot \mathbf{v}_2}{\|\mathbf{v}_1\| \|\mathbf{v}_2\|} \right|\]

Tip

We use the absolute value of the dot product to ensure we get the acute angle. If we didn’t, \(\cos \theta\) could be negative, giving an obtuse angle.

The angle between lines is typically considered the acute angle, hence the absolute value. This formula directly leverages the dot product’s ability to relate vector orientation to a scalar value.

Distance from a Point to a Line

The distance from a point \(S\) to a line \(\ell\) (passing through point \(P\) and parallel to vector \(\mathbf{v}\)) is:

\[d = \frac{\|\overrightarrow{PS} \times \mathbf{v}\|}{\|\mathbf{v}\|} = \|\overrightarrow{PS} \times \hat{\mathbf{v}}\|\] where \(\overrightarrow{PS}\) is the vector from any point \(P\) on the line to the point \(S\), and \(\hat{\mathbf{v}}\) is the unit direction vector of the line.

Distance Point to Line

This formula elegantly uses the cross product. The magnitude of the cross product of \(\overrightarrow{PS}\) and \(\mathbf{v}\) is \(\|\overrightarrow{PS}\| \|\mathbf{v}\| \sin \theta\), where \(\theta\) is the angle between \(\overrightarrow{PS}\) and \(\mathbf{v}\). Dividing by \(\|\mathbf{v}\|\) gives \(\|\overrightarrow{PS}\| \sin \theta\), which is exactly the perpendicular distance from S to the line.

Planes in 3-Space: Vector Equation

A plane \(M\) in space is defined by:

• A point on the plane \(P_0\) with position vector \(\mathbf{r}_0\).
• A non-zero normal vector \(\mathbf{n}\) that is perpendicular to the plane.

For any point \(P\) with position vector \(\mathbf{r}\) on the plane, the vector \(\overrightarrow{P_0P}\) lies in the plane and is therefore perpendicular to \(\mathbf{n}\).

So, \(\mathbf{n} \cdot \overrightarrow{P_0P} = 0\), which can be written as: \[\mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0\] Rearranging, we get the vector equation of the plane: \[\mathbf{r} \cdot \mathbf{n} = \mathbf{r}_0 \cdot \mathbf{n} = d\]

Planes in 3-Space: Vector Equation

Plane Definition

The normal vector is the key to defining a plane. It tells us the orientation of the plane in space. The vector equation states that the projection of any vector from the origin to a point on the plane onto the normal vector is a constant, ‘d’.

Planes in 3-Space: Scalar Equation

Given \(P_0 = (x_0, y_0, z_0)\) and \(\mathbf{n} = (a, b, c)\), and \(P = (x, y, z)\):

From \(\mathbf{n} \cdot \overrightarrow{P_0P} = 0\): \[\left( \begin{array}{c} a \\ b \\ c \end{array} \right) \cdot \left( \begin{array}{c} x - x_0 \\ y - y_0 \\ z - z_0 \end{array} \right) = 0\] Expanding this gives the scalar equation of the plane: \[a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\] This can be simplified to the general form: \[ax + by + cz = d\] where \(d = ax_0 + by_0 + cz_0\).

The scalar equation of a plane is a linear equation in x, y, and z. The coefficients a, b, and c are directly the components of the normal vector. This form is often the most convenient for algebraic manipulations and solving systems of equations.

Distance from a Point to a Plane

The distance from a point \(S\) to a plane (with normal vector \(\mathbf{n}\) and containing point \(P\)) is the length of the vector projection of \(\overrightarrow{PS}\) onto \(\mathbf{n}\):

\[d = \left| \frac{\overrightarrow{PS} \cdot \mathbf{n}}{\|\mathbf{n}\|} \right| = |\overrightarrow{PS} \cdot \hat{\mathbf{n}}|\]

Alternatively, if the plane is \(ax + by + cz = d_{plane}\) and the point is \(S(x_S, y_S, z_S)\), the distance is:

\[D = \frac{|ax_S + by_S + cz_S - d_{plane}|}{\sqrt{a^2 + b^2 + c^2}}\]

Distance from a Point to a Plane

Distance Point to Plane

Similar to the distance from a point to a line, this formula uses projections. The absolute value ensures the distance is positive. The second formula is a direct application of the first one, converting the dot product into component form and simplifying. It is very handy for quick calculations when the plane equation is given in general form.

Angles Between Two Planes

The angle between two planes is defined as the acute angle between their respective normal vectors.

If plane 1 has normal \(\mathbf{n}_1\) and plane 2 has normal \(\mathbf{n}_2\), and \(\theta\) is the acute angle between the planes:

\[\cos \theta = \left| \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|} \right|\] Therefore: \[\theta = \cos^{-1} \left( \left| \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|} \right| \right)\]

Just like lines, the angle between planes is generally taken to be the acute angle. This is why we use the absolute value of the dot product, ensuring that \(\cos \theta\) is non-negative and \(\theta\) falls within \([0, \pi/2]\). This is a direct application of the angle formula for vectors.

Key Takeaways

1. Vectors have Magnitude & Direction: They represent physical quantities like displacement, velocity, force.
2. Geometric & Algebraic Representation: Vectors can be visualized as directed segments and calculated using components in coordinate systems.
3. Vector Operations:
   • Addition/Subtraction: Head-to-tail rule, parallelogram rule (component-wise).
   • Scalar Multiplication: Changes magnitude and/or reverses direction (component-wise).
4. Dot Product:
   • Result: Scalar.
   • Applications: Angle between vectors, work done, projection length, checking for perpendicularity.
5. Cross Product (3D only):
   • Result: Vector perpendicular to both input vectors.
   • Applications: Finding normal vectors, area of parallelogram/triangle.
6. Scalar Triple Product (3D only):
   • Result: Scalar.
   • Applications: Volume of parallelepiped, checking for coplanarity.
7. Lines & Planes:
   • Lines: Defined by a point and a direction vector.
   • Planes: Defined by a point and a normal vector.
   • Equations: Vector, Parametric, Cartesian forms for lines; Vector, Scalar forms for planes.
   • Applications: Distance calculations, angles between geometric objects.

This slide summarizes the most critical concepts from the chapter. Make sure you understand each of these points. They are the building blocks for more advanced topics in calculus, linear algebra, and various engineering disciplines.

Key Equations

Equation	Description
\(\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{v}\)	Vector equation of a line
\(\frac{x - x_0}{v_1} = \frac{y - y_0}{v_2} = \frac{z - z_0}{v_3}\)	Cartesian equation of a line
\(\mathbf{r} \cdot \mathbf{n} = d\)	Vector equation of a plane
\(ax + by + cz = d\)	Scalar equation of a plane
\(\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2}\)	Length (magnitude) of a vector
\(\hat{\mathbf{u}} = \frac{1}{\|\mathbf{u}\|} \mathbf{u}\)	Unit vector in direction of \(\mathbf{u}\)
\(\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta\)	Dot product (geometric definition)
\(\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3\)	Dot product (component form)
\(\mathbf{u} \times \mathbf{v} = (\|\mathbf{u}\| \|\mathbf{v}\| \sin \theta) \hat{\mathbf{n}}\)	Cross product (geometric definition)
\(\mathbf{u} \times \mathbf{v} = \left| \begin{array}{c c c} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{array} \right|\)	Cross product (determinant form)
\(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \left| \begin{array}{c c c} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{array} \right|\)	Scalar triple product
\(\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2}\right) \mathbf{v}\)	Vector projection of \(\mathbf{u}\) onto \(\mathbf{v}\)
\(W = \mathbf{F} \cdot \mathbf{D}\)	Work done by constant force \(\mathbf{F}\) through displacement \(\mathbf{D}\)
\(D = \frac{|ax_S + by_S + cz_S - d_{plane}|}{\sqrt{a^2 + b^2 + c^2}}\)	Distance from point \(S(x_S, y_S, z_S)\) to plane \(ax+by+cz=d_{plane}\)

This table is a concise reference for all the major formulas introduced in the chapter. Familiarize yourself with these equations and understand when and how to apply each one.

Key Terms

Term	Definition
Vector	A quantity having both magnitude and direction.
Magnitude (Norm)	The length of a vector.
Direction	The orientation of a vector in space.
Zero Vector	A vector with zero magnitude and no specific direction (\(\mathbf{0}\)).
Unit Vector	A vector with a magnitude of 1.
Scalar	A quantity having only magnitude (e.g., mass, temperature, time).
Component Form	Representing a vector by its projections along coordinate axes (e.g., \((u_1, u_2)\)).
Dot Product	A scalar product of two vectors, related to the cosine of the angle between them.
Cross Product	A vector product of two vectors in \(\mathbb{R}^3\), perpendicular to both.
Scalar Triple Product	The dot product of one vector with the cross product of two others.
Coplanar Vectors	Vectors that lie in the same plane.
Normal Vector	A vector perpendicular to a plane.
Direction Vector	A vector parallel to a line, indicating its orientation.
Projection	The component of one vector along the direction of another.

Understanding these terms is essential for clear communication and comprehension of vector concepts. Make sure you can define each term and use it correctly in context.