Express the vector as a product of its length and direction.
-$\frac { 8 } { 3 } \mathbf { j } - 2 \mathbf { k }$
A) $\frac { 10 } { 3 } ( \mathrm { j } - \mathbf { k } )$
B) $\frac { 100 } { 9 } \left( \frac { 4 } { 5 } \mathrm { j } - \frac { 3 } { 5 } \mathbf { k } \right)$
C) $\frac { 10 } { 3 } \left( \frac { 4 } { 5 } \mathrm { j } - \frac { 3 } { 5 } \mathbf { k } \right)$
D) $\frac { 10 } { 3 } \left( \frac { 8 } { 3 } \mathbf { j } - 2 \mathbf { k } \right)$

Question

Quizplus · Accepted Answer

To find the length of the vector, we use the formula $\|\mathbf{v}\|=\sqrt{v_x^2+v_y^2+v_z^2}$ where $\mathbf{v}$ is the given vector. Plugging in the values, we get $\|\mathbf{v}\|=\sqrt{0^2+\left(\frac{8}{3}\right)^2+(-2)^2}=\frac{10}{3}$. To find the direction of the vector, we divide each component by the length: $$\frac{1}{\|\mathbf{v}\|}\left(\frac{8}{3}\mathbf{j}-2\mathbf{k}\right)=\frac{10}{3}\left(\frac{4}{5}\mathbf{j}-\frac{3}{5}\mathbf{k}\right)$$ Therefore, the answer is choice C.

Express the Vector as a Product of Its Length and Direction