# Magnitudes of Scalar Multiples

Lesson

Having looked at magnitudes of vectors, and scalar multiples of vectors we can combine these to find the magnitude of scalar multiples.

represents the magnitude of . We can calculate it using Pythagoras' theorem.

A scalar multiple is found by extending the length of by a multiple, let's call $k$k.  So it makes sense then that the magnitude of the scalar multiple is a scalar multiple of the magnitude.

If the vector was defined using column matrix notation, then

##### Example 1

Find the magnitude of the vector pictured, after it is scaled by a multiple of $3$3

Method 1 - Find the magnitude then scale it by 3.

Magnitude of $u$u

$\left|u\right|=\sqrt{2^2+\left(-6\right)^2}=\sqrt{40}=2\sqrt{10}$|u|=22+(6)2=40=210

Then scale by $3$3, so magnitude of $3u$3u

$3\times\left|u\right|=3\times2\sqrt{10}=6\sqrt{10}$3×|u|=3×210=610

Method 2 - Find the new scaled vector, then find that magnitude.

$3u$3u will be 

Now find the magnitude of this

$\sqrt{6^2+\left(-18\right)^2}=\sqrt{360}=6\sqrt{10}$62+(18)2=360=610

Both answers yield the same result.