Chapter 11.2, Problem 70E

a.

To determine

Find $u\cdot w$ and $v\cdot w$ .

Answer to Problem 70E

  $\boxed{\begin{array}{l}u\cdot w=0\\ v\cdot w=0\end{array}}$

Explanation of Solution

Given information:

Use the figure below.

  

Find $u\cdot w$ and $v\cdot w$ .

Calculation:

Consider the given diagram which shows vector $u,v,w$ .

  

From the diagram $u=\left(0,2,0,\right)$ and $w=\left(0,0,3\right)$ as initial point are origin itself.

If vector $v$ represented by the directed line from $P\left[p_1,p_2,p_3\right]$ and $Q\left[q_1,q_2,q_3\right]$ then the component form of $v$ is obtained by subtracting coordinates as follows,

  $v=\left(q_1-p_1,q_2-p_2,q_3-p_3\right)$

Let $p\left(0,4,0,\right)$ and $q\left(0,2,0\right)$

  $\begin{array}{l}v=\left(0-0,2-4,0-0\right)\\ =0,-2,0\end{array}$

Hence component from vector $v$ is $\boxed{v=0,-2,0}$

Now find the dot product of vector $u=\left(0,2,0,\right)$ and $w=\left(0,0,3\right)$ .

  $\begin{array}{l}u\cdot w=\langle 0,2,0\rangle \cdot \langle 0,0,3\rangle \\ =0\cdot 0+2\cdot 0+0\cdot 3\\ =0+0+0\\ =\boxed{0}\end{array}$

Now find the dot product of vector $v=\left(0,-2,0\right)$ and $w=\left(0,0,3\right)$ .

  $\begin{array}{l}v\cdot w=\langle 0,-2,0\rangle \cdot \langle 0,0,3\rangle \\ =0\cdot 0+\left(-2\right)\cdot 0+0\cdot 3\\ =0+0+0\\ =\boxed{0}\end{array}$

Hence, $\boxed{\begin{array}{l}u\cdot w=0\\ v\cdot w=0\end{array}}$

b.

To determine

Find $u\cdot w$ and $v\cdot w$ .

Answer to Problem 70E

  $\boxed{\begin{array}{l}u\cdot w=0\\ v\cdot w=0\end{array}}$

Explanation of Solution

Given information:

Use the figure below.

  

A vector $z$ is parallel to $u$ and $v$ .What is the dot product of $z$ and $w$ explains your reasoning.

Calculation Consider the result of part (a) if two vectors are perpendicular then dot product is zero, so vector $w$ is perpendicular to both vectors $u$ and $v$ .

Vector $z$ is parallel to $u$ and $v$ , then it is perpendicular to $w$ therefore dot product of $z$ and $w$ will also be zero.

Hence $\boxed{z\cdot w=0}$