# Which vectors are perpendicular to each other

## Vectors orthogonal

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5th grade (Austrian school plan)

**Definition:**

Two vectors stand

**orthogonal**on top of each other if the two vectors have a

**right angle**lock in.

Example:

**How do you check whether two vectors are orthogonal to each other?**

Calculate the scalar product of the two vectors.

If the scalar product results in 0, then the two vectors stand **at a right angle** on each other.

Example:

If the vectors $ \ vec {a} = \ begin {pmatrix} 4 \ 5 \ \ end {pmatrix} $ and $ \ vec {b} = \ begin {pmatrix} 10 \ -8 \ \ end { pmatrix} $ at right angles to each other?**solution**

$ \ vec {a} \ cdot \ vec {b} = \ begin {pmatrix} 4 \ 5 \ \ end {pmatrix} \ cdot \ begin {pmatrix} 10 \ -8 \ \ end {pmatrix} = 4 \ times 10 + 5 \ times (-8) = 40 + (-40) = 0 $

Since the scalar product is 0, the vectors $ \ vec {a} = \ begin {pmatrix} 4 \ 5 \ \ end {pmatrix} $ and $ \ vec {b} = \ begin {pmatrix} 10 \ - 8 \ \ end {pmatrix} $ at right angles to each other.

**How do you find a normal vector for a given vector?**

For a vector there is **always** two associated normal vectors:

For the left-handed normal vector you swap the x-coordinate with the y-coordinate and then change the sign of the x-coordinate.

As a formula: The vector $ \ begin {pmatrix} a_1 \ a_2 \ \ end {pmatrix} $ becomes $ \ begin {pmatrix} -a_2 \ a_1 \ \ end {pmatrix} $.

For the normal vector rotated to the right, you swap the x-coordinate with the y-coordinate and then change the sign of the y-coordinate.

As a formula: The vector $ \ begin {pmatrix} a_1 \ a_2 \ \ end {pmatrix} $ becomes $ \ begin {pmatrix} a_2 \ -a_1 \ \ end {pmatrix} $.

Example:

Find an orthogonal vector to $ \ vec {a} = \ begin {pmatrix} 2 \ 3 \ \ end {pmatrix} $ **solution**

Swapping the coordinates and changing the sign of the x-coordinate results in $ \ vec {a_ {L}} = \ begin {pmatrix} -3 \ 2 \ \ end {pmatrix} $

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