The Cross Product

A product yielding a vector perpendicular to both factors, whose magnitude reads as the area of the parallelogram the factors span.

Active Calculus Multivariable, The Cross Product

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8 exercises

Activity 9.5.2

Suppose $\vu = \langle 0, 1, 3\rangle$ and $\vv = \langle 2, -1, 0\rangle$ . Use equation this equation for the following.

Find the cross product $\vu\times\vv$ .
Evaluate the dot products $\vu\cdot(\vu\times\vv)$ and $\vv\cdot(\vu\times\vv)$ . What does this tell you about the geometric relationship among $\vu$ , $\vv$ , and $\vu\times\vv$ ?
Find the cross product $\vv\times \vi$ .
Recall that multiplication of real numbers is associative. For example, $(2\cdot 5)\cdot 3 = 2\cdot(5\cdot 3)$ . Is it true that the cross product of vectors is associative? For instance, is it true that $(\vu\times\vv)\times\vi = \vu\times(\vv\times\vi)$ ?
Find the c…

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Activity 9.5.3

Find the area of the parallelogram formed by the vectors $\vu = \langle 1,3, -2\rangle$ and $\vv=\langle 3,0,1\rangle$ .
Find the area of the parallelogram in $\R^3$ whose vertices are $(1,0,1)$ , $(0,0,1)$ , $(2,1,0)$ , and $(1,1,0)$ .

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Activity 9.5.5

In this activity, we are focused on the type of objects being used and whether the expression makes sense to do at all. We are not going to worry about interpreting or understanding what is being measured by these expressions.

For each of the expressions below, state whether the result is a scalar, a vector, or undefined. You should write a sentence or two about each to explain your reasoning. (Ass…

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Exercise 18

Let $\vu = 2\vi + \vj$ and $\vv = \vi + 2\vj$ be vectors in $\R^3$ .

Without doing any computations, find a unit vector that is orthogonal to both $\vu$ and $\vv$ . What does this tell you about the formula for $\vu \times \vv$ ?

Using the properties of the cross product and what you know about cross products involving the fundamental vectors $\vi$ and $\vj$ , compute $\vu \times \vv$ .

Next, use the determinant version of Equation this equation to compute $\vu \times \vv$ . Write one sentence that compa…

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Exercise 19

Let $\vx = \langle 1, 1, 1 \rangle$ and $\vy = \langle 0, 3, -2 \rangle$ .

Are $\vx$ and $\vy$ orthogonal? Are $\vx$ and $\vy$ parallel? Clearly explain how you know, using appropriate vector products.
Find a unit vector that is orthogonal to both $\vx$ and $\vy$ .
Express $\vy$ as the sum of two vectors: one parallel to $\vx$ , the other orthogonal to $\vx$ .
Determine the area of the parallelogram formed by $\vx$ and $\vy$ .

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Exercise 20

Consider the triangle in $\R^3$ formed by $P=(3, 2, -1)$ , $Q=(1, -2, 4)$ , and $R=(4, 4, 0)$ .

Find $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ .
Observe that the area of $\triangle PQR$ is half of the area of the parallelogram formed by $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ . Hence find the area of $\triangle PQR$ .
Find a unit vector that is orthogonal to the plane that contains points $P$ , $Q$ , and $R$ .
Determine the measure of $\angle QPR$ .

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Exercise 21

One of the properties of the cross product is that $(\vu+\vv) \times \vw = (\vu \times \vw) + (\vv \times \vw)$ . That is, the cross product distributes over vector addition on the right. Here we investigate whether the cross product distributes over vector addition on the left.

Let $\vu = \langle 1,2,-1 \rangle$ , $\vv = \langle 4,-3,6 \rangle$ , and $\vw = \langle 4,7,2 \rangle$ . Calculate What do you notice?
Use the properties of the cross product to show that in general for any vectors $\vx$ , $\vy$ , and $\vz$ in $\R^3$ .

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Exercise 23

In The Vector Nature of the Cross Product, we stated “Because the cross product works well with linear combinations, $\vu\times\vv$ is perpendicular to any linear combination of $\vu$ and $\vv$ .” We want to verify this algebraically here. Specifically, show that $\vu\times\vv$ will be orthogonal to any vector of the form $a\vu+b\vv$ .

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Adapted from Active Calculus Multivariable © 2013–2026 Nicholas Long, Mitchel T. Keller, and Steven Schlicker. Licensed under CC BY-SA 4.0. Source: Active Calculus Multivariable. Imported at revision 692c03c.