Magnitudes of Scalar Multiples

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=2√10

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

$3\times\left|u\right|=3\times2\sqrt{10}=6\sqrt{10}$3×|u|=3×2√10=6√10

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=6√10

Both answers yield the same result.