# How do I find the unit vector for v = < 2,-5,6 >?

Aug 10, 2014

A unit vector just means a vector whose length equals 1 unit. We want a unit vector u in the v -direction. We will use u = v/|v|.

Note: Any vector parallel to v can be written as c v with real c; if c > 0 this goes in the same direction as v.

The length |v| of a vector v = < x,y,z > is

| v | = $\sqrt{{x}^{2} + {y}^{2} + {z}^{2}} ,$ and so

|< 2, -5, 6 >| $= \sqrt{{2}^{2} + {\left(- 5\right)}^{2} + {6}^{2}}$

$= \sqrt{4 + 25 + 36} = \sqrt{65} ,$

we can divide v by $\sqrt{65}$ to get

$u = < \frac{2}{\sqrt{65}} , \frac{- 5}{\sqrt{65}} , \frac{6}{\sqrt{65}} >$

Feel free to approximate this with decimals if asked, but this is the exact "math" answer. dansmath to the rescue!