Find a unit vector in the direction of a given vector

Question

Find a unit vector in the direction of $[1,2,3]$.

Are we just trying to find the norm or length of the vector?

$\sqrt{(1^2)+(2^2)+(3^3)} = \sqrt{14}$

Thank you :)

Answer 1

Let $v = (1,2,3)$. The length of $v$ is, as you found, $\|v\| = \sqrt{14}$. Scaling it down by its own length, will give you the appropriate unit vector of same direction :

$$v' = \frac{1}{\|v\|}v = \frac{1}{\sqrt{14}}(1,2,3) = \bigg(\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}\bigg)$$

Cross validate the result now :

$$\|v'\| = \Bigg\|\bigg(\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}}\bigg)\Bigg\| = \sqrt{\bigg(\frac{1}{\sqrt{14}}\bigg)^2 + \bigg(\frac{2}{\sqrt{14}}\bigg)^2 + \bigg(\frac{3}{\sqrt{14}}\bigg)^2}$$ $$=$$ $$\sqrt{\frac{1}{14} + \frac{4}{14} + \frac{9}{14}}=\sqrt{\frac{14}{14}} = 1$$