Chapter 12.1, Problem 62E

To determine

To calculate: The inner product of $\overrightarrow{u}=\langle 2,-1,3\rangle$ and $\overrightarrow{v}=\langle 5,3,0\rangle$ .

Answer to Problem 62E

The inner product is $\overrightarrow{u}\cdot \overrightarrow{v}=7$ .

Explanation of Solution

Given information:

The vectors $\overrightarrow{u}=\langle 2,-1,3\rangle$ and $\overrightarrow{v}=\langle 5,3,0\rangle$ .

Formula used:

The inner product of two vectors $\overrightarrow{u}=\langle a_1,b_1,c_1\rangle$ and $\overrightarrow{v}=\langle a_2,b_2,c_2\rangle$ is expressed as $\overrightarrow{u}\cdot \overrightarrow{v}=a_1{a}_2+b_1{b}_2+c_1{c}_2$ .

Calculation:

Consider the vectors $\overrightarrow{u}=\langle 2,-1,3\rangle$ and $\overrightarrow{v}=\langle 5,3,0\rangle$ .

Recall that the inner product of two vectors $\overrightarrow{u}=\langle a_1,b_1,c_1\rangle$ and $\overrightarrow{v}=\langle a_2,b_2,c_2\rangle$ is expressed as $\overrightarrow{u}\cdot \overrightarrow{v}=a_1{a}_2+b_1{b}_2+c_1{c}_2$ .

Apply it,

$\begin{array}{c}\overrightarrow{u}\cdot \overrightarrow{v}=2\left(5\right)+\left(-1\right)\left(3\right)+3\left(0\right)\\ =10-3+0\\ =7\end{array}$

Thus, the inner product is $\overrightarrow{u}\cdot \overrightarrow{v}=7$ .