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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


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.

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