What is scaling in math

Scaling vectors

The multiplication of a vector by a number is called scaling a vector. The product is in turn a vector that corresponds to the value multiplied by it

  • longer $ \ longrightarrow (2 \ cdot \ vec {a}) $,
  • shorter $ \ longrightarrow (0.5 \ cdot \ vec {a}) $ or even
  • in the opposite direction $ \ longrightarrow (-0.5 \ cdot \ vec {a}) $

is re-mapped.

 

In the graphic above you can clearly see that a vector $ \ vec {a} $ multiplied by a scalargreater $ 1 $ (e.g. $ 2 $) becomes longer.

If, on the other hand, a vector is multiplied by a scalar between $ 0 $ and $ -1 $, this is shortened and its direction also changes by 180 °.

When a vector is multiplied by a scalar between $ 0 $ and $ 1 $, the length of the vector is shortened, but its direction remains the same.

When multiplying by a scalar less than $ -1 $, the vector is lengthened and its direction changes by 180 °. The vector is then drawn in exactly the opposite way.

Example: scaling vectors

Click here to expand

We consider the vector $ \ vec {a} = \ left (\ begin {array} {c} 4 \ 6 \ end {array} \ right) $.

Calculate:

a) $ 2.5 \ vec {a} $

b) $ -1.25 \ vec {a} $

c) $ 0.75 \ vec {a} $

d) $ -0.5 \ vec {a} $

a)

$ 2.5 \ vec {a} = 2.5 \ cdot \ left (\ begin {array} {c} 4 \ 6 \ end {array} \ right) = \ left (\ begin {array} {c} 2 , 5 \ times 4 \ 2.5 \ times 6 \ end {array} \ right) = \ left (\ begin {array} {c} 10 \ 15 \ end {array} \ right) $

The output vector elongates and maintains its direction.

b)

$ -1.25 \ vec {a} = -1.25 \ cdot \ left (\ begin {array} {c} 4 \ 6 \ end {array} \ right) = \ left (\ begin {array} { c} -1.25 \ times 4 \ -1.25 \ times 6 \ end {array} \ right) = \ left (\ begin {array} {c} -5 \ -7.5 \ end {array } \ right) $

The output vector lengthens and changes its direction by 180 °.

c)

$ 0.75 \ vec {a} = 0.75 \ cdot \ left (\ begin {array} {c} 4 \ 6 \ end {array} \ right) = \ left (\ begin {array} {c} 0 , 75 \ cdot 4 \ 0.75 \ cdot 6 \ end {array} \ right) = \ left (\ begin {array} {c} 3 \ 4.5 \ end {array} \ right) $

The output vector shortens and maintains its direction.

d)

$ -0.5 \ vec {a} = -0.5 \ cdot \ left (\ begin {array} {c} 4 \ 6 \ end {array} \ right) = \ left (\ begin {array} { c} -0.5 \ cdot 4 \ -0.5 \ cdot 6 \ end {array} \ right) = \ left (\ begin {array} {c} -2 \ -3 \ end {array} \ right) $

The output vector is shortened by half and changes its direction by 180 °.