# 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

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 °.