# Orthogonal vectors

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**vectors**a and b are

**orthogonal**, if their dot product is equal to zero.

## Examples of tasks

### Examples of plane tasks

In the case of the plane problem for the vectors a = {a_{x}; a_{y}} and b = {b_{x}; b_{y}} **orthogonality condition** can be written by the following formula:

_{x}· b

_{x}+ a

_{y}· b

_{y}= 0

**Solution:**

Calculate the dot product of these vectors:

a · b = 1 · 2 + 2 · (-1) = 2 - 2 = 0**Answer:** since the dot product is zero, the vectors a and b are orthogonal.

**Solution:**

Calculate the dot product of these vectors:

a · b = 3 · 7 + (-1) · 5 = 21 - 5 = 16**Answer:** since the dot product is not zero, the vectors a and b are not orthogonal.

**Solution:**

Calculate the dot product of these vectors:

a · b = 2 · n + 4 · 1 = 2n + 42n + 4 = 0

2n = -4

n = -2

**Answer:** vectors a and b are orthogonal when n = -2.

### Examples of spatial tasks

In the case of the plane problem for the vectors a = {a_{x}; a_{y}; a_{z}} and b = {b_{x}; b_{y}; b_{z}} **orthogonality condition** can be written by the following formula:

_{x}· b

_{x}+ a

_{y}· b

_{y}+ a

_{z}· b

_{z}= 0

**Solution:**

Calculate the dot product of these vectors:

a · b = 1 · 2 + 2 · (-1) + 0 · 10 = 2 - 2 + 0 = 0**Answer:** since the dot product is zero, the vectors a and b are orthogonal.

**Solution:**

Calculate the dot product of these vectors:

a · b = 2 · 3 + 3 · 1 + 1 · (-9) = 6 + 3 -9 = 0**Answer:** since the dot product is zero, the vectors a and b are orthogonal.

**Solution:**

Calculate the dot product of these vectors:

a · b = 2 · n + 4 · 1 + 1 · (-8)= 2n + 4 - 8 = 2n - 42n - 4 = 0

2n = 4

n = 2

**Answer:** vectors a and b are orthogonal when n = 2.

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