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
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
Then scale by $3$3, so magnitude of $3u$3u
Method 2 - Find the new scaled vector, then find that magnitude.
$3u$3u will be $$
Now find the magnitude of this
Both answers yield the same result.