Which vectors are perpendicular to each other

Vectors orthogonal

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5th grade (Austrian school plan)


Definition:
Two vectors stand orthogonalon top of each other if the two vectors have a right anglelock 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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