Express the vector as a product of its length and direction.
-$$- 2 \mathbf { i } - 8 \mathbf { j } + \frac { 8 } { 3 } \mathbf { k }$$
A) $\frac { 26 } { 3 } ( - \mathbf { i } - \mathrm { j } + \mathbf { k } )$
B) $\frac { 26 } { 3 } \left( - \frac { 3 } { 13 } \mathbf { i } - \frac { 12 } { 13 } \mathbf { j } + \frac { 4 } { 13 } \mathbf { k } \right)$
C) $\frac { 676 } { 9 } \left( - \frac { 3 } { 13 } \mathbf { i } - \frac { 12 } { 13 } \mathbf { j } + \frac { 4 } { 13 } \mathbf { k } \right)$
D) $\frac { 3 } { 26 }$

Question

Quizplus · Accepted Answer

To express a vector in terms of its length and direction, we need to find the unit vector in the direction of the given vector and then multiply it by the magnitude of the given vector.

The magnitude of the given vector is: 
$$
\begin{aligned} 
\left\| - 2 \mathbf { i } - 8 \mathbf { j } + \frac { 8 } { 3 } \mathbf { k } ight\| &= \sqrt{(-2)^2 + (-8)^2 + \left(\frac{8}{3}ight)^2} \ 
&= \sqrt{4 + 64 + \frac{64}{9}} = \frac{26}{3}
\end{aligned}
$$

The unit vector in the direction of the given vector is obtained by dividing the given vector by its magnitude: 
$$
\begin{aligned} 
\mathbf{u} &= \frac{- 2 \mathbf { i } - 8 \mathbf { j } + \frac { 8 } { 3 } \mathbf { k }}{\frac{26}{3}} \ 
&= \left(-\frac{3}{13}ight) \mathbf{i} - \left(\frac{12}{13}ight) \mathbf{j} + \left(\frac{4}{13}ight) \mathbf{k} 
\end{aligned}
$$

Therefore, the given vector as a product of its length and direction is: 
$$
\frac { 26 } { 3 } \left( - \frac { 3 } { 13 } \mathbf { i } - \frac { 12 } { 13 } \mathbf { j } + \frac { 4 } { 13 } \mathbf { k } ight)
$$

The answer is B.

Express the Vector as a Product of Its Length and Direction