Question

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

Answer 1

The normalized vector is `(2/3i+1/3j+2/3k)`.

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 `(ai+bj+ck)` 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) = sqrt9=3`

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

The normalized vector is `(2/3i+1/3j+2/3k)`.

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