Linear Algebra

3.5 Cross Product

Imron Rosyadi

The Cross Product

A Vector Operation in 3-Space

Introduction to the Cross Product

The cross product is a vector operation unique to 3-dimensional space. Unlike the dot product, which results in a scalar, the cross product produces a vector. It’s crucial for understanding concepts like torque, angular momentum, and magnetic forces in physics and engineering.

In ECE, particularly in electromagnetism, mechanics, and computer graphics, the cross product is indispensable. It allows us to find a vector perpendicular to two given vectors, which is fundamental for defining normal vectors to surfaces, calculating torque, or determining the direction of magnetic fields. While the dot product measures the “alignment” of two vectors, the cross product measures their “perpendicularity” and gives a direction.

Definition of the Cross Product

If \(\mathbf{u} = (u_{1}, u_{2}, u_{3})\) and \(\mathbf{v} = (v_{1}, v_{2}, v_{3})\) are vectors in 3-space, then the cross product \(\mathbf{u} \times \mathbf{v}\) is defined as:

\[ \mathbf{u} \times \mathbf{v} = (u_{2}v_{3} - u_{3}v_{2}, u_{3}v_{1} - u_{1}v_{3}, u_{1}v_{2} - u_{2}v_{1}) \]

Determinant Notation

This can be remembered using a determinant format:

\[ \mathbf{u} \times \mathbf{v} = \biggl( \left| \begin{array}{cc} u_{2} & u_{3} \\ v_{2} & v_{3} \end{array} \right|, - \left| \begin{array}{cc} u_{1} & u_{3} \\ v_{1} & v_{3} \end{array} \right|, \left| \begin{array}{cc} u_{1} & u_{2} \\ v_{1} & v_{2} \end{array} \right| \biggr) \quad \text{(1)} \]

Tip

Mnemonic for Calculation: Form a \(2 \times 3\) matrix: \(\left[ \begin{array}{lll}u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \end{array} \right]\)

• 1st Component: Delete 1st column, take determinant.
• 2nd Component: Delete 2nd column, take negative of determinant.
• 3rd Component: Delete 3rd column, take determinant.

The cross product is only defined for vectors in 3-space. This definition might look complicated, but the determinant mnemonic makes it much easier to recall. The resulting vector is perpendicular to both u and v, a property we will explore further. In computer graphics, if u and v define two edges of a polygon, their cross product gives the normal vector to that polygon, which is essential for lighting calculations and surface rendering.

Example 1: Calculating a Cross Product

Find \(\mathbf{u} \times \mathbf{v}\), where \(\mathbf{u} = (1, 2, -2)\) and \(\mathbf{v} = (3, 0, 1)\).

Solution:

Using the determinant notation or mnemonic:

\[ {\begin{array}{r l}&{\mathbf{u}\times\mathbf{v}={\bigg(}{\bigg|}{\begin{array}{l l}{2}&{-2}\\ {0}&{1}\end{array}}{\bigg|},-{\bigg|}{\begin{array}{l l}{1}&{-2}\\ {3}&{1}\end{array}}{\bigg|},{\bigg|}{\begin{array}{l l}{1}&{2}\\ {3}&{0}\end{array}}{\bigg|}{\bigg)}}\\ &{\qquad=(2,-7,-6)}\end{array}} \]

Example 1: Calculating a Cross Product

Interactive Cross Product Calculator

Enter the components of \(\mathbf{u}\) and \(\mathbf{v}\) to calculate their cross product.

This interactive tool allows you to quickly compute cross products. Try changing the input vectors. For instance, what happens if u and v are parallel? (e.g., u = [1,2,3], v = [2,4,6]). The result should be the zero vector, as parallel vectors do not define a unique perpendicular direction. This is a direct application of numpy.cross, a highly optimized function in Python’s numerical library.

Properties of Cross Product: Theorem 3.5.1

The cross product has fundamental relationships with the dot product and important geometric implications.

If \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in 3-space:

1. \(\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) = 0\) \([\mathbf{u} \times \mathbf{v}\) is orthogonal to \(\mathbf{u}]\)
2. \(\mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) = 0\) \([\mathbf{u} \times \mathbf{v}\) is orthogonal to \(\mathbf{v}]\)
3. \(\| \mathbf{u} \times \mathbf{v}\| ^2 = \| \mathbf{u}\| ^2 \| \mathbf{v}\| ^2 - (\mathbf{u} \cdot \mathbf{v})^2\) [Lagrange’s identity]
4. \(\mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\) [Vector Triple Product]
5. \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{v} \cdot \mathbf{w}) \mathbf{u}\) [Vector Triple Product]

Important

Properties (a) and (b) are key: The cross product of two vectors is always orthogonal (perpendicular) to both original vectors. This is its defining geometric characteristic.

The orthogonality property is incredibly useful. For example, in electromagnetism, the force on a charge moving through a magnetic field (Lorentz force) is given by a cross product, and the force is always perpendicular to both the velocity and the magnetic field. Lagrange’s identity (c) provides a link between the magnitudes of the cross product and dot product, which we’ll use for geometric interpretations. The vector triple product formulas (d) and (e) are important for simplifying complex vector expressions, especially in advanced mechanics or field theory.

Interactive: Orthogonality Verification

Let’s verify properties (a) and (b) using the vectors from Example 1.

\(\mathbf{u} = (1, 2, -2)\), \(\mathbf{v} = (3, 0, 1)\), and \(\mathbf{u} \times \mathbf{v} = (2, -7, -6)\).

This interactive check confirms that the dot product of the cross product with each of the original vectors is indeed zero, demonstrating their orthogonality. This property is fundamental in physics, for instance, when calculating the normal to a surface or the direction of magnetic fields. In computer graphics, surface normals, derived from cross products, are used to determine how light reflects off objects.

Properties of Cross Product: Theorem 3.5.2

The cross product also has several algebraic properties.

If \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are any vectors in 3-space and \(k\) is any scalar:

1. \(\mathbf{u}\times \mathbf{v} = -(\mathbf{v}\times \mathbf{u})\) (Anti-commutative)
2. \(\mathbf{u}\times (\mathbf{v} + \mathbf{w}) = (\mathbf{u}\times \mathbf{v}) + (\mathbf{u}\times \mathbf{w})\) (Distributive over vector addition)
3. \((\mathbf{u} + \mathbf{v})\times \mathbf{w} = (\mathbf{u}\times \mathbf{w}) + (\mathbf{v}\times \mathbf{w})\) (Distributive over vector addition)
4. \(k(\mathbf{u}\times \mathbf{v}) = (k\mathbf{u})\times \mathbf{v} = \mathbf{u}\times (k\mathbf{v})\) (Scalar multiplication)
5. \(\mathbf{u}\times \mathbf{0} = \mathbf{0}\times \mathbf{u} = \mathbf{0}\) (Zero vector property)
6. \(\mathbf{u}\times \mathbf{u} = \mathbf{0}\) (Self cross product)

The anti-commutative property (a) is important: the order of vectors matters, and reversing it flips the direction of the resulting vector. This is unlike scalar multiplication or dot product. Properties (b), (c), and (d) show that the cross product interacts well with vector addition and scalar multiplication, making it a linear operation in each argument. Property (f) implies that the cross product of a vector with itself (or with a parallel vector) is always the zero vector, as the vectors do not define a unique perpendicular direction.

Cross Products of Standard Unit Vectors

The standard unit vectors are \(\mathbf{i} = (1,0,0)\), \(\mathbf{j} = (0,1,0)\), \(\mathbf{k} = (0,0,1)\).

Calculations show:

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

And due to anti-commutativity:

• \(\mathbf{j}\times \mathbf{i} = -\mathbf{k}\)
• \(\mathbf{k}\times \mathbf{j} = -\mathbf{i}\)
• \(\mathbf{i}\times \mathbf{k} = -\mathbf{j}\)

Also, for self-cross products: - \(\mathbf{i}\times \mathbf{i} = \mathbf{0}\) - \(\mathbf{j}\times \mathbf{j} = \mathbf{0}\) - \(\mathbf{k}\times \mathbf{k} = \mathbf{0}\)

Figure 3.5.2: Mnemonic for unit vector cross products.

This diagram is a fantastic mnemonic! Moving clockwise (i to j to k to i) gives the next vector. Moving counter-clockwise gives the negative of the next vector. This pattern directly reflects the right-hand rule, which is a convention for determining the direction of the cross product, vital in electromagnetism for current, magnetic fields, and forces.

Determinant Form of Cross Product

The cross product can also be expressed symbolically using a \(3 \times 3\) determinant.

\[ \mathbf{u}\times \mathbf{v} = \left| \begin{array}{lll}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ u_{1} & u_{2} & u_{3}\\ v_{1} & v_{2} & v_{3} \end{array} \right| = \left| \begin{array}{lll}u_{2} & u_{3}\\ v_{2} & v_{3} \end{array} \right|\mathbf{i} - \left| \begin{array}{lll}u_{1} & u_{3}\\ v_{1} & v_{3} \end{array} \right|\mathbf{j} + \left| \begin{array}{lll}u_{1} & u_{2}\\ v_{1} & v_{2} \end{array} \right|\mathbf{k} \quad \text{(4)} \]

Example: \(\mathbf{u} = (1,2,-2)\), \(\mathbf{v} = (3,0,1)\)

\[ \mathbf{u}\times \mathbf{v} = \left| \begin{array}{lll}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ 1 & 2 & -2\\ 3 & 0 & 1 \end{array} \right| = (2)(1) - (-2)(0)\mathbf{i} - ((1)(1) - (-2)(3))\mathbf{j} + ((1)(0) - (2)(3))\mathbf{k} = 2\mathbf{i} - 7\mathbf{j} - 6\mathbf{k} \]

Warning

The cross product is not associative! \(\mathbf{u}\times (\mathbf{v}\times \mathbf{w}) \neq (\mathbf{u}\times \mathbf{v})\times \mathbf{w}\) For example, \(\mathbf{i}\times (\mathbf{j}\times \mathbf{j}) = \mathbf{0}\), but \((\mathbf{i}\times \mathbf{j})\times \mathbf{j} = -\mathbf{i}\).

This \(3 \times 3\) determinant form is perhaps the most common way to calculate a cross product by hand, as it naturally produces the components in terms of i, j, and k. It’s essentially a compact way of writing the component-wise definition. The non-associativity warning is critical! This is a common pitfall. The order of operations matters significantly, just as it does for matrix multiplication.

Geometric Interpretation: The Right-Hand Rule

The direction of \(\mathbf{u} \times \mathbf{v}\) for non-zero vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by the right-hand rule.

• Curl the fingers of your right hand from \(\mathbf{u}\) towards \(\mathbf{v}\) through the smaller angle between them.
• Your thumb will point in the direction of \(\mathbf{u} \times \mathbf{v}\).

This rule is a convention that establishes the orientation of the resulting vector in 3-space.

Figure 3.5.3: The Right-Hand Rule.

The right-hand rule is more than just a convention; it’s deeply embedded in physics and engineering. It’s used to determine the direction of magnetic fields around currents, the direction of torque, or the direction of propagation of electromagnetic waves. It’s a visual and intuitive way to understand the vector direction that results from a cross product. Practice with i x j = k to get a feel for it.

Geometric Interpretation: Area of a Parallelogram

The magnitude of the cross product has a direct geometric meaning.

From Lagrange’s identity: \(\| \mathbf{u}\times \mathbf{v}\|^{2} = \| \mathbf{u}\|^{2}\| \mathbf{v}\|^{2} - (\mathbf{u}\cdot \mathbf{v})^{2}\). Using \(\mathbf{u}\cdot \mathbf{v} = \| \mathbf{u}\| \| \mathbf{v}\| \cos \theta\): \[ \| \mathbf{u}\times \mathbf{v}\| = \| \mathbf{u}\| \| \mathbf{v}\| \sin \theta \quad \text{(6)} \] This formula is precisely the area of the parallelogram determined by vectors \(\mathbf{u}\) and \(\mathbf{v}\).

Theorem 3.5.3: If \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in 3-space, then \(\| \mathbf{u}\times \mathbf{v}\|\) is equal to the area of the parallelogram determined by \(\mathbf{u}\) and \(\mathbf{v}\).

Figure 3.5.4: Area of parallelogram.

This is a beautiful geometric interpretation. The sin(theta) factor naturally accounts for how “spread out” the vectors are. If they are parallel (\(\theta=0\) or \(\theta=\pi\)), then \(\sin\theta=0\), and the area (and cross product magnitude) is zero, which makes sense as parallel vectors don’t form a parallelogram with positive area. This is very useful in computer graphics for calculating surface areas or in physics for magnetic flux calculations.

Example 4: Area of a Triangle

Find the area of the triangle determined by points \(P_1(2,2,0)\), \(P_2(-1,0,2)\), and \(P_3(0,4,3)\).

1. Form vectors: \(\overrightarrow{P_1P_2} = (-1-2, 0-2, 2-0) = (-3, -2, 2)\) \(\overrightarrow{P_1P_3} = (0-2, 4-2, 3-0) = (-2, 2, 3)\)

2. Calculate cross product: \(\overrightarrow{P_1P_2} \times \overrightarrow{P_1P_3} = ((-2)(3) - (2)(2), (2)(-2) - (-3)(3), (-3)(2) - (-2)(-2))\) \(= (-6-4, -4+9, -6-4) = (-10, 5, -10)\)

3. Calculate magnitude: \(\| \overrightarrow{P_1P_2} \times \overrightarrow{P_1P_3}\| = \sqrt{(-10)^2 + 5^2 + (-10)^2} = \sqrt{100 + 25 + 100} = \sqrt{225} = 15\)

4. Area of triangle: \(A = \frac{1}{2} \| \overrightarrow{P_1P_2} \times \overrightarrow{P_1P_3}\| = \frac{1}{2} (15) = \frac{15}{2}\)

Figure 3.5.5: Triangle formed by three points.

The area of a triangle formed by three points is simply half the area of the parallelogram formed by two vectors originating from one of those points. This is a practical application of the cross product. In computer graphics, this is used to calculate the area of triangular facets, which are the basic building blocks of 3D models.

Interactive: Area of Triangle Calculator

Enter the coordinates of three points to calculate the area of the triangle they form.

This interactive tool automates the process of finding the area of a triangle in 3D space. It reinforces the understanding that the cross product’s magnitude is geometrically significant. This can be used in surveying, architectural design, or in robotics for path planning where areas of regions need to be computed.

Scalar Triple Product

The scalar triple product involves three vectors and results in a scalar.

Definition 2: If \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) are vectors in 3-space, then \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\) is called the scalar triple product.

It can be calculated using a \(3 \times 3\) determinant: \[ \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \left| \begin{array}{ccc}u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{array} \right| \quad \text{(7)} \]

Example 5: Calculation

For \(\mathbf{u} = (3, -2, -5)\), \(\mathbf{v} = (1, 4, -4)\), \(\mathbf{w} = (0, 3, 2)\):

\[ \begin{array}{r l} & {\mathbf{u}\cdot (\mathbf{v}\times \mathbf{w}) = \left| \begin{array}{l l l}{3} & {-2} & {-5}\\ {1} & {4} & {-4}\\ {0} & {3} & {2} \end{array} \right|}\\ & {\qquad = 3(4 \cdot 2 - (-4) \cdot 3) - (-2)(1 \cdot 2 - (-4) \cdot 0) + (-5)(1 \cdot 3 - 4 \cdot 0)}\\ & {\qquad = 3(8 + 12) + 2(2) - 5(3)}\\ & {\qquad = 3(20) + 4 - 15 = 60 + 4 - 15 = 49} \end{array} \]

The scalar triple product is a very efficient way to compute this specific combination of dot and cross products. The order of operations is fixed by the parentheses: v x w is calculated first, resulting in a vector, then dotted with u. The determinant form simplifies the calculation significantly. Note that writing \(\mathbf{u} \cdot \mathbf{v} \times \mathbf{w}\) without parentheses is usually understood to mean \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\) because \((\mathbf{u} \cdot \mathbf{v}) \times \mathbf{w}\) is not a valid operation (you can’t cross a scalar with a vector).

Geometric Interpretation of Determinants

The absolute value of determinants has profound geometric meaning.

Theorem 3.5.4 (a): The absolute value of \(\operatorname{det}\left[ \begin{smallmatrix}u_{1} & u_{2} \\ v_{1} & v_{2} \end{smallmatrix} \right]\) is the area of the parallelogram in 2-space determined by \(\mathbf{u}=(u_1, u_2)\) and \(\mathbf{v}=(v_1, v_2)\).

Theorem 3.5.4 (b): The absolute value of \(\operatorname{det}\left[ \begin{smallmatrix}u_{1} & u_{2} & u_{3}\\ v_{1} & v_{2} & v_{3}\\ w_{1} & w_{2} & w_{3} \end{smallmatrix} \right]\) is the volume of the parallelepiped in 3-space determined by \(\mathbf{u}, \mathbf{v}, \mathbf{w}\). \[ V = |\mathbf{u}\cdot (\mathbf{v}\times \mathbf{w})| \quad \text{(9)} \]

Figure 3.5.7: (a) Area of 2D parallelogram, (b) Volume of 3D parallelepiped.

This theorem connects determinants, a purely algebraic concept, to fundamental geometric quantities: area and volume. The 2D case can be seen as a special case of the cross product’s magnitude by embedding the vectors in the XY-plane of 3D space. The 3D case directly relates to the scalar triple product. This is incredibly important in fields like numerical analysis and computational geometry, where areas and volumes need to be calculated efficiently, for example, in finite element analysis or fluid dynamics simulations.

Interactive: Volume of Parallelepiped Calculator

Enter the components of three vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) to calculate the volume of the parallelepiped they determine.

This tool computes the volume using the scalar triple product. Try making the vectors coplanar (e.g., w = u + v or a linear combination of u and v). What do you expect the volume to be? If the vectors are coplanar, they will not form a 3D parallelepiped, and the volume should be zero. This leads directly into our next concept: the test for coplanarity.

Coplanarity Test: Theorem 3.5.5

The scalar triple product provides a simple test for whether three vectors lie in the same plane.

If the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) have the same initial point, then they lie in the same plane if and only if:

\[ \mathbf{u}\cdot (\mathbf{v}\times \mathbf{w}) = \left| \begin{array}{lll}u_{1} & u_{2} & u_{3}\\ v_{1} & v_{2} & v_{3}\\ w_{1} & w_{2} & w_{3} \end{array} \right| = 0 \]

Note

Coplanar points: A set of points is coplanar if you can draw a single flat surface (a plane) that contains all of them.

Coplanar vectors: Vectors are coplanar if their linear combinations lie within the same plane.

Geometrically, if three vectors are coplanar, they cannot form a parallelepiped with a positive volume. Hence, their scalar triple product (which is the signed volume) must be zero.

This theorem is extremely useful in geometry and physics. For example, in structural engineering, determining if three forces are coplanar is crucial for analyzing equilibrium. In computer graphics, it can be used to check if three points lie on the same plane, which is important for rendering and collision detection. It’s a powerful and elegant consequence of the geometric interpretation of the scalar triple product.

Conclusion: Key Takeaways

• Definition: The cross product \(\mathbf{u} \times \mathbf{v}\) yields a vector in 3-space orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).
• Properties: It’s anti-commutative, distributive, but NOT associative.
• Geometric Magnitude: \(\| \mathbf{u} \times \mathbf{v}\|\) represents the area of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\).
• Direction: Determined by the right-hand rule.
• Scalar Triple Product: \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\) represents the signed volume of the parallelepiped formed by \(\mathbf{u}, \mathbf{v}, \mathbf{w}\).
• Coplanarity Test: Three vectors are coplanar if their scalar triple product is zero.

Important

The cross product and scalar triple product are fundamental tools in ECE for understanding 3D geometry, forces, fields, and spatial relationships.

Today, we’ve explored the cross product, a unique and powerful operation in 3D vector algebra. Its ability to generate a perpendicular vector, its connection to areas and volumes, and its role in determining coplanarity are all incredibly valuable. These concepts are not just theoretical; they are the backbone of many practical applications in electromagnetism, robotics, computer graphics, and mechanics. Keep these geometric interpretations in mind as you encounter more complex problems in your ECE studies.