# How do you normalize  (2i+j+2k) ?

Jan 31, 2016

The normalized vector is $\left(\frac{2}{3} i + \frac{1}{3} j + \frac{2}{3} k\right)$.

This is a unit vector in the same direction as the original vector.

#### Explanation:

Normalizing a vector involves dividing the coefficient of each of its elements by the length of the vector, to yield a unit vector (length =1) in the same direction as the original vector.

To do this we first need to find the length of the vector. If the vector is in the form $\left(a i + b j + c k\right)$ then its length is given by:

$l = \sqrt{{a}^{2} + {b}^{2} + {c}^{2}}$

In this case this is:

$l = \sqrt{{2}^{2} + {1}^{2} + {2}^{2}} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$

Now we divide $a$, $b$ and $c$ by $l$, which is $3$:

The normalized vector is $\left(\frac{2}{3} i + \frac{1}{3} j + \frac{2}{3} k\right)$.

This is a unit vector in the same direction as the original vector.