Chapter 9.4, Problem 49E

(a)

To determine

Whether the given vectors $\text{u}$ and $\text{v}$ are parallel.

Answer to Problem 49E

The given vectors $\text{u}$ and $\text{v}$ are parallel.

Explanation of Solution

Given information:

The given vectors are $\text{u}=\langle 3,-2,4\rangle \text{,}\text{v=}\langle -6,4,-8\rangle$ .

Calculation:

The two vectors are parallel one vector must be a scalar multiple of the other.

Multiply the first vector by $-2$ .

$\begin{array}{c}\left(-2\right)\text{u}=\left(-2\right)\left(3,-2,4\right)\\ -2\text{u}=\left(-6,4,-8\right)\\ \text{v}=\left(-6,4,-8\right)\end{array}$

From the above equations $\text{u,}\text{v}$ are parallel.

Therefore, the given vectors $\text{u}$ and $\text{v}$ are parallel.

(b)

To determine

Whether the given vectors $\text{u}$ and $\text{v}$ are parallel.

Answer to Problem 49E

The given vectors $\text{u}$ and $\text{v}$ are parallel.

Explanation of Solution

Given information:

The given vectors are $\text{u}=\langle -9,-6,12\rangle \text{,}\text{v=}\langle -12,8,-16\rangle$ .

Calculation:

The two vectors are parallel one vector must be a scalar multiple of the other.

Multiply the first vector by $-4/3$ .

$\begin{array}{c}-\frac{4}{3}\text{u}=-\frac{4}{3}\left(-9,-6,12\right)\\ -\frac{4}{3}\text{u}=\left(12,8,-16\right)\\ \text{v}=\left(12,8,-16\right)\end{array}$

From the above equations $\text{u,}\text{v}$ are parallel.

Therefore, the given vectors $\text{u}$ and $\text{v}$ are parallel.

(c)

To determine

Whether the given vectors $\text{u}$ and $\text{v}$ are parallel.

Answer to Problem 49E

The given vectors $\text{u}$ and $\text{v}$ are not parallel.

Explanation of Solution

Given information:

The given vectors are $\text{u}=\text{i}+\text{j}+\text{k,}\text{v}=2\text{i}+2\text{j}-\text{2k}$ .

Calculation:

Rewrite the given vectors.

$\begin{array}{l}\text{u}=\left(1,1,1\right)\\ \text{v}=\left(2,2,-2\right)\end{array}$

The two vectors are parallel one vector must be a scalar multiple of the other.

Multiply the first vector by $2$ .

$\begin{array}{c}\left(2\right)\text{u}=\left(2\right)\left(1,1,1\right)\\ 2\text{u}=\left(2,2,2\right)\\ \text{v}=\left(2,2,-2\right)\end{array}$

From the above equations, no scalar can be found such that $\text{u,}\text{v}$ are the same so they are not are parallel.

Therefore, the given vectors $\text{u}$ and $\text{v}$ are not parallel.