# 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 scalar**greater** $ 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 °.

- How much is 5 5 4
- What is meant by license fees
- What assumptions does theology make
- Can grief turn someone into a psychopath
- What are the motives behind imperialism
- Why are music videos so monetizable?
- How can you prove 2 1
- What are the types of money markets
- What is the allure of ax throwing
- How do I join Autism Speaks
- Can someone become like an ESTJ
- Why is the human cycle being recycled
- What are the methods for adult education
- What kind of plant is the 5th
- Why is middleware important
- Why do we have ethanol gasoline
- What is the name of a social security application
- What color of hibiscus do you like
- EMDR therapy is a form of hypnosis
- Where is the Teslas Gigafactory located
- Noida Ghaziabad is more developed than Lucknow
- Why is Great Britain doing nothing today?
- How does light come from the sun
- What is a two-color injection machine