Chapter 11.2, Problem 15E

a.

To determine

To sketch:A vector $2\overrightarrow{v}$ for the given vector $\overrightarrow{v}$ .

Explanation of Solution

Given:

The given vector is $\overrightarrow{v}=\left(1,1,3\right)$ .

Formula/ concept used:

The vector $\overrightarrow{OP}=\overrightarrow{r}=\left(a,b,c\right)$ is a vector whose initial point is at the origin and the tip is at point $P\left(1,1,3\right)$ and components along X-, Y-, and Z-axis are a , b , and crespectively .

Calculations:

The given vector is $\overrightarrow{v}=\left(1,1,3\right)$ . Therefore, vector $2\overrightarrow{v}=\left(2,2,6\right)$ .

The vector $\overrightarrow{OP}=2\overrightarrow{v}=\left(2,2,6\right)$ or $\overrightarrow{OP}=2\overrightarrow{v}=\text{2i+2j+6k}$ is shown below.

  

b.

To determine

To sketch: A vector $-\overrightarrow{v}$ for the given vector $\overrightarrow{v}$ .

Explanation of Solution

Given:

The given vector is $\overrightarrow{v}=\left(1,1,3\right)$ .

Formula/ concept used:

The vector $\overrightarrow{OP}=\overrightarrow{r}=\left(a,b,c\right)$ is a vector whose initial point is at the origin and the tip is at point $P\left(1,1,3\right)$ and components along X-, Y-, and Z-axis are a , b , and crespectively .

Calculations:

The given vector is $\overrightarrow{v}=\left(1,1,3\right)$ . Therefore, vector $-\overrightarrow{v}=\left(-1,-1,-3\right)$ .

The vector $\overrightarrow{OP}=-\overrightarrow{v}=\left(-1,-1,-3\right)$ or $\overrightarrow{OP}=-\overrightarrow{v}=\text{-i-j-k}$ is shown in Figure-2 here.

  

C.

To determine

To sketch: A vector $\frac{3}{2}\overrightarrow{v}$ for the given vector $\overrightarrow{v}$ .

Explanation of Solution

Given:

The given vector is $\overrightarrow{v}=\left(1,1,3\right)$ .

Formula/ concept used:

The vector $\overrightarrow{OP}=\overrightarrow{r}=\left(a,b,c\right)$ is a vector whose initial point is at the origin and the tip is at point $P\left(1,1,3\right)$ and components along X-, Y-, and Z-axis are a , b , and crespectively .

Calculations:

The given vector is $\overrightarrow{v}=\left(1,1,3\right)$ . Therefore, the vector $\frac{3}{2}\overrightarrow{v}=\frac{3}{2}\left(1,1,3\right)=\left(\mathrm{1.51.5},4.5\right)$ .

The vector $\overrightarrow{OP}=\frac{3}{2}\overrightarrow{v}=\left(1.5,1.5,4.5\right)$ or $\overrightarrow{OP}=\frac{3}{2}\overrightarrow{v}=\text{1}\text{.5i+1}\text{.5j+4}\text{.5k}$ is shown in Figure-3 here.

  

d.

To determine

To sketch: A vector $-\frac{1}{2}\overrightarrow{v}$ for the given vector $\overrightarrow{v}$ .

Explanation of Solution

Given:

The given vector is $\overrightarrow{v}=\left(1,1,3\right)$ .

Formula/ concept used:

The vector $\overrightarrow{OP}=\overrightarrow{r}=\left(a,b,c\right)$ is a vector whose initial point is at the origin and the tip is at point $P\left(1,1,3\right)$ and components along X-, Y-, and Z-axis are a , b , and crespectively .

Calculations:

The given vector is $\overrightarrow{v}=\left(1,1,3\right)$ . Therefore, the vector $-\frac{1}{2}\overrightarrow{v}=-\frac{1}{2}\left(1,1,3\right)=\left(-0.5,-0.5,-1.5\right)$ .

The vector $\overrightarrow{OP}=-\frac{1}{2}\overrightarrow{v}=\left(-0.5,-0.5,-1.5\right)$ or $\overrightarrow{OP}=-\frac{1}{2}\overrightarrow{v}=-\text{0}\text{.5i-0}\text{.5j-1}\text{.5k}$ is shown in Figure-4 here.