Question

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

$v=2i-j+k$

Short answer

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

Step-by-step solution

Step 1. Given

The vector $v=2i-j+k$

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,

$$\begin{gathered} ||v||=||2i-j+k|| \\ =\sqrt{2^2+{(-1)}^2+1^2} \\ =\sqrt{4+1+1} \\ =\sqrt{6} \end{gathered}$$

Step 4. Find the unit vector.

The unit vector is,

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