1.5 The Cross Product

The Cross Product

Introduction

Computing the cross product

The Vector Nature of the Cross Product

Applications of the Cross Product

Comparing the dot and cross products

Our first task is to understand what it means to complete a right-handed coordinate system. For each of the cases below, you need to give a vector that fills in the blank to create a right-handed coordinate system. For example, the answer that would complete $\{\vi, \vj, ???\}$ would be $\vk$. You should be careful to when you have vectors that are in the opposite directions of $\vi$ or $\vj$ or $\vk$. -$\{\vj,\vi, ???\}$-$\{-\vi,\vj, ???\}$-$\{\vk,\vi, ???\}$-$\{-\vi,-\vj, ???\}$-$\{\vk,\vj, ???\}$

Work through this exercise and explain your reasoning step by step.

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 cross product $\vu\times\vu$ and write a sentence or two to explain the meaning of your result.

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 $(1,1,0)$.

Find two unit vectors orthogonal to both $\vu$ and $\vv$.

Find the volume of the parallelepiped formed by the vectors $\vu$, $\vv$, and $\vw = \langle 3,3,1\rangle$.

Find a vector orthogonal to the parallelogram containing the points $(0,1,2)$, $(4,1,0)$, and $(-2,2,2)$.

Given the vectors $\vu$ and $\vv$ shown below in Figure, sketch the cross products $\vu\times\vv$ and $\vv\times\vu$.

Are the vectors $\va = \langle 1,3,-2\rangle$, $\vb=\langle2,1,-4\rangle$, and $\vc=\langle 0, 1, 0\rangle$ in standard position coplanar? Use the concepts from this section to explain your answer.