Advanced Vector Calculus: Background You’ll Need 1

  • Calculate the cross product of two vectors and understand what it tells us

The Cross Product

The cross product multiplies two vectors in [latex]\mathbb{R}^3[/latex] to produce a third vector that is orthogonal to both original vectors. This operation is essential for finding perpendicular directions, calculating torque, and determining areas and volumes.

cross product

For vectors [latex]\mathbf{u} = \langle u_1, u_2, u_3 \rangle[/latex] and [latex]\mathbf{v} = \langle v_1, v_2, v_3 \rangle[/latex]:

 

Component Form:

[latex]\mathbf{u} \times \mathbf{v} = \langle u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1 \rangle[/latex]

Standard Unit Vector Form:

[latex]\mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2)\mathbf{i} - (u_1v_3 - u_3v_1)\mathbf{j} + (u_1v_2 - u_2v_1)\mathbf{k}[/latex]

Direction: The Right-Hand Rule

The cross product formula tells us the components of [latex]\mathbf{u} \times \mathbf{v}[/latex], but how do we determine its direction? The right-hand rule provides the answer.

Right-Hand Rule Steps:

  1. Point your right hand fingers in the direction of the first vector [latex]\mathbf{u}[/latex]
  2. Curl your fingers toward the second vector [latex]\mathbf{v}[/latex]
  3. Your thumb points in the direction of [latex]\mathbf{u} \times \mathbf{v}[/latex]
This figure has two images. The first image has three vectors with the same initial point. Two of the vectors are labeled “u” and “v.” The angle between u and v is theta. The third vector is perpendicular to u and v. It is labeled “u cross v.” The second image has three vectors. The vectors are labeled “u, v, and u cross v.” “u cross v” is perpendicular to u and v. Also, on the image of these three vectors is a right hand. The fingers are in the direction of u. As the hand is closing, the direction of the closing fingers is the direction of v. The thumb is up and in the direction of “u cross v.”
Figure 2. The direction of [latex]{\bf{u}}\times{\bf{v}}[/latex] is determined by the right-hand rule.
Order Matters: Try the right-hand rule for [latex]\mathbf{v} \times \mathbf{u}[/latex]. Start with fingers pointing toward [latex]\mathbf{v}[/latex], curl toward [latex]\mathbf{u}[/latex]—your thumb points in the opposite direction! This shows that [latex]\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})[/latex].
Let [latex]\mathbf{u} =\langle 0,2,1 \rangle[/latex] and [latex]\mathbf{v} =\langle 3,-1,0 \rangle[/latex]. Calculate [latex]\mathbf{u} \times \mathbf{v}[/latex] and [latex]\mathbf{v} \times \mathbf{u}[/latex] and graph them."This

Cross Products of Standard Unit Vectors

unit vector cross products

Positive Results (right-hand rule gives positive direction):

  • [latex]\mathbf{i} \times \mathbf{j} = \mathbf{k}[/latex]
  • [latex]\mathbf{j} \times \mathbf{k} = \mathbf{i}[/latex]
  • [latex]\mathbf{k} \times \mathbf{i} = \mathbf{j}[/latex]

Negative Results (opposite order gives negative direction):

  • [latex]\mathbf{j} \times \mathbf{i} = -\mathbf{k}[/latex]
  • [latex]\mathbf{k} \times \mathbf{j} = -\mathbf{i}[/latex]
  • [latex]\mathbf{i} \times \mathbf{k} = -\mathbf{j}[/latex]
Notice the cyclic pattern [latex]\mathbf{i} \to \mathbf{j} \to \mathbf{k} \to \mathbf{i}[/latex]. Going forward in the cycle gives positive results, backward gives negative results.
Find [latex]\mathbf{i} \times (\mathbf{j} \times \mathbf{k})[/latex].

Properties of the Cross Product

The dot product and cross product are both forms of vector multiplication, but they have very different properties and applications. The dot product is called the scalar product because it produces a scalar, while the cross product is called the vector product because it produces a vector.

properties of the cross product:

Let [latex]\mathbf{u}[/latex], [latex]\mathbf{v}[/latex], and [latex]\mathbf{w}[/latex] be vectors, and let [latex]c[/latex] be a scalar.

  1. Anticommutative: [latex]\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})[/latex]
  2. Distributive: [latex]\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}[/latex]
  3. Scalar multiplication: [latex]c(\mathbf{u} \times \mathbf{v}) = (c\mathbf{u}) \times \mathbf{v} = \mathbf{u} \times (c\mathbf{v})[/latex]
  4. Zero vector: [latex]\mathbf{u} \times \mathbf{0} = \mathbf{0} \times \mathbf{u} = \mathbf{0}[/latex]
  5. Self cross product: [latex]\mathbf{v} \times \mathbf{v} = \mathbf{0}[/latex]
  6. Scalar triple product: [latex]\mathbf{u} \times (\mathbf{v} \times \mathbf{w}) \neq (\mathbf{u} \times \mathbf{v}) \times \mathbf{w}[/latex] (in general)
Use the cross product properties to calculate [latex](2\mathbf{i} \times 3\mathbf{j}) \times \mathbf{j}[/latex].

Use the properties of the cross product to find the magnitude of the cross product of [latex]\mathbf{u} = \langle 0,4,0\rangle[/latex] and [latex]\mathbf{v} = \langle 0,0,-3\rangle[/latex].

Magnitude of the Cross Product

We’ve focused on the direction of [latex]\mathbf{u} \times \mathbf{v}[/latex], but what about its magnitude? It turns out there is a simple expression for the magnitude of [latex]\mathbf{u} \times \mathbf{v}[/latex] involving the magnitudes of [latex]\mathbf u[/latex] and [latex]\mathbf v[/latex], and the sine of the angle between them.

magnitude of the cross product

For vectors [latex]\mathbf{u}[/latex] and [latex]\mathbf{v}[/latex] with angle [latex]\theta[/latex] between them:

[latex]||\mathbf{u} \times \mathbf{v}|| = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \sin \theta[/latex]

Compare this to the dot product formula [latex]\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos \theta[/latex]. The cross product uses sine instead of cosine!

The cross product only works in three dimensions. In 2D, there’s no direction simultaneously perpendicular to two nonparallel vectors—you’d need to point “out of the plane,” which requires a third dimension. This geometric requirement explains why cross products are fundamental in 3D applications like torque, rotation, and magnetic fields, but don’t exist in 2D vector algebra.