The Cross Product

Section2.4The Cross Product

ObjectivesTopical

perform the cross-product of vectors

The dot product gave us a way of multiplying two vectors together, but the result was a number, not a vectors. We now define the cross product, which will allow us to multiply two vectors together to give us another vector. We were able to define the dot product in all dimensions. The cross product is only defined in \(\mathbb{R}^3\text{.}\)

Definition2.4.1The Cross Product

The cross product of two vectors \(\vec u = \left\lt u_1,u_2,u_3\right>\) and \(\vec v = \left\lt v_1,v_2,v_3\right>\) is a new vector \(\vec u\times \vec v\text{.}\) This new vector is (1) orthogonal to both \(\vec u\) and \(\vec v\text{,}\) (2) has a length equal to the area of the parallelogram whose sides are these two vectors, and (3) points in the direction your thumb points as you curl the base of your right hand from \(\vec u\) to \(\vec v\text{.}\) The formula for the cross product is

\begin{equation*} \vec u\times \vec v = \left\lt u_2v_3-u_3v_2,-(u_1v_3-u_3v_1),u_1v_2-u_2v_1\right> = \det\begin{bmatrix}\vec i \amp \vec j\amp \vec k\\ u_1\amp u_2\amp u_3\\ v_1\amp v_2\amp v_3 \end{bmatrix} . \end{equation*}

This definition is not really a definition. It is actually a theorem. If you use the formula given as the definition, then you would need to prove the three facts. We have the tools to give a complete proof of (1) and (3), but we would need a course in linear algebra to prove (2). It shouldn't be too much of a surprise that the cross product is related to area, since it is defined in terms of determinants

Subsection2.4.1Basics with the Cross Product

Exercise2.4.1

Let \(\vec u=\langle 1,-2,3\rangle\) and \(\vec v=\langle 2,0,-1\rangle\text{.}\)

(a)

Compute \(\vec u\times \vec v\) and \(\vec v\times \vec u\text{.}\) How are they related?

(b)

Compute \(\vec u \cdot (\vec u\times \vec v)\) and \(\vec v \cdot (\vec u\times \vec v)\text{.}\) Explain why you get the answer you got?

(c)

Compute \(\vec u \times (2\vec u)\text{.}\) Explain why you get the answer you got?

Exercise2.4.2

Consider the vectors \({\ii}=(1,0,0)\text{,}\) \({2\jj}=(0,2,0)\text{,}\) and \({3\kk}=(0,0,3)\text{.}\)

(a)

Compute \(\ii\times {2\jj}\) and \({2\jj}\times {\ii}\text{.}\)

(b)

Compute \({\ii}\times {3\kk}\) and \({3\kk}\times {\ii}\text{.}\)

(c)

Compute \({2\jj}\times {3\kk}\) and \({3\kk}\times {2\jj}\text{.}\)

Give a geometric reason as to why some vectors above have a plus sign, and some have a minus sign.

Exercise2.4.3

Let \(P=(2,0,0)\text{,}\) \(Q=(0,3,0)\text{,}\) and \(R=(0,0,4)\text{.}\) Find a vector that orthogonal to both \(\vec {PQ}\) and \(\vec {PR}\text{.}\) Then find the area of the triangle \(PQR\text{.}\) Construct a 3D graph of this triangle.

Hint

What shape does two triangles side-by-side make?

Subsection2.4.2Computational Practice

These are provided to help you achieve better skills in finding basic computational answers