Question

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

$v=3i-6j-2k$

Short answer

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

Step-by-step solution

Step 1. Given

The vector $v=3i-6j-2k$

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,

$$\begin{gathered} ||v||=||3i-6j-2k|| \\ =\sqrt{3^2+{(-6)}^2+{(-2)}^2} \\ =\sqrt{9+36+4} \\ =\sqrt{49} \\ =7 \end{gathered}$$

Step 4. Find the unit vector.

The unit vector is,

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