Question

In Problems 45–50, find the unit vector in the same direction as v.

$v=-6i+12j+4k$

Short answer

The unit vector in the same direction as $v=-6i+12j+4k$is $-\frac{3}{7}i+\frac{6}{7}j+\frac{2}{7}k$

Step-by-step solution

Step 1. Given

The vector $v=-6i+12j+4k$

To find the unit vector.

Step 2. Unit vector in the direction of v 

For any nonzero vector v, the vector $u=\frac{v}{||v||}$is a unit vector that has the same direction as v.

Step 3. Find v

Now,

$$\begin{gathered} ||v||=||-6i+12j+4k|| \\ =\sqrt{{(-6)}^2+{\left(12\right)}^2+4^2} \\ =\sqrt{36+144+16} \\ =\sqrt{196} \\ =14 \end{gathered}$$

Step 4. Find the unit vector.

The unit vector is,

$$\begin{gathered} u=\frac{v}{||v||} \\ =\frac{-6i+12j+4k}{14} \\ =-\frac{3}{7}i+\frac{6}{7}j+\frac{2}{7}k \end{gathered}$$