Question

Find the unit vector that has the same direction as the vector v.v=

Answer 1

Solution:

Given a vector:

$v=-6i-8j$

The unit vector is expressed as

$\begin{gathered} \frac{\vec{v}}{\lvert v\rvert} \\ \text{where} \\ \lvert v\rvert\text{ is the magnitude }of\text{ the vector} \end{gathered}$

step 1: Evaluate the magnitude of the vector.

The magnitude of vector v is evaluated as

$\begin{gathered} |v|=\sqrt[]{(-6)^2+(-8)^2} \\ =\sqrt[]{36+64} \\ =\sqrt[]{100} \\ \Rightarrow|v|=10 \end{gathered}$

step 2: Evaluate the unit vector.

$\begin{gathered} \frac{\vec{v}}{\lvert v\rvert}=\frac{1}{|v|}\times\vec{v} \\ =\frac{1}{10}(-6i-8j) \\ =-\frac{6}{10}i-\frac{8}{10}j \\ \Rightarrow-\frac{3}{5}i-\frac{4}{5}j \end{gathered}$

Hence, the unit vector that has the same direction as the vector v

is

$-\frac{3}{5}i-\frac{4}{5}j$