Chapter 10.2, Problem 40E

To determine

To find the dot product of $\text{u}$ and $\text{v}$ .

Answer to Problem 40E

The dot product of $\text{u}$ and $\text{v}$ is $-42$ .

Explanation of Solution

Given information:

  $\text{u}=6i+8j+4k$ and $\text{v}=-i-6j+3k$ .

The dot product between unit vectors:

  $i\cdot j=i\cdot k=j\cdot k=0$ and $i\cdot i=j\cdot j=k\cdot k=1$ .

Let, $\text{a}$ and $\text{b}$ are two vectors.

Where, $\text{a}=(a_1,a_2,a_3)=a_{1}i+a_{2}j+a_{3}k$ and $\text{b}=(b_1,b_2,b_3)=b_{1}i+b_{2}j+b_{3}k$ .

  $\begin{array}{l}\therefore \text{a}\cdot \text{b}=(a_{1}i+a_{2}j+a_{3}k)\cdot (b_{1}i+b_{2}j+b_{3}k)\\ =a_1{b}_1+a_2{b}_2+a_3{b}_3\end{array}$

Here,

  $(a_1,a_2,a_3)=(6,8,4)$ and $(b_1,b_2,b_3)=(-1,-6,3)$ .

  $\begin{array}{l}\therefore \text{u}\cdot \text{v}=(6i+8j+4k)\cdot (-i-6j+3k)\\ =\left\{6\times (-1)\right\}+\left\{8\times (-6)\right\}+(4\times 3)\\ =-6+(-48)+12\\ =-6-48+12\\ =-42\end{array}$

Hence,

The dot product of $\text{u}$ and $\text{v}$ is $-42$ .