In this page, you will learn about the result of vector dot product of orthogonal vectors.

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When are two vectors called 'orthogonal' vectors? `color(coral)(text(orth)) + color(deepskyblue)(text(gonia))` means `color(coral)(text(right))+color(deepskyblue)(text(angled))`.

- have `90^@` angle between them
- perpendicular to each other
- the vectors are right-angled
- all the above
- all the above

The answer is 'All the above'

Given the definition of dot product as

`vec p cdot vec q = |vec p||vec q|cos theta`

What is `vec p cdot vec q`, when the given vectors are orthogonal?

- `|p||q|cos 90`
- `|p||q| 0`
- `0`
- all the above
- all the above

The answer is 'All the above'.

Given two non-zero vectors `vec p, vec q in bbb V` have `vec p cdot vec q = 0`, then what is the angle between the vectors?

- `0^@`
- `90^@`
- `90^@`
- `180^@`
- Any one of the above

The answer is '`90^@`'. It can either be `90^@` or `270^@`.

• Dot product of orthogonal vectors is `0`.

If dot product of two non-zero vectors is `0`, then the vectors are orthogonal.

**Dot Product of Orthogonal Vectors: ** For any pair of orthogonal vectors `vec p, vec q in bbb V`,

`vec p cdot vec q = 0`

**Angle between vectors when dot product is `0`:** For any pair of non-zero vectors `vec p, vec q in bbb V`, If `vec p cdot vec q = 0` then the vectors are orthogonal. The angle between them is `+- 90^@`.

*Solved Exercise Problem: *

Given `vec p= 2i+j-k` and `vec q = i+2j+4k` what is the angle between them?

- `90^@`
- `90^@`
- `0`
- `180^@`
- `0^@`

The answer is '`90^@`'. As the dot product between the vectors is `0`.

*Solved Exercise Problem: *

Given two vectors `vec p` and `vec q` in x-y plane. They make angles `13^@` and `-77^@` with x-axis. What is `vec p cdot vec q`?

- `90`
- `0`
- `0`
- `13xx77`
- `13xx(-77)`

The answer is '`0`'. The vectors are orthogonal as the angle between them is `13+77`. Note that the vectors are in x-y plane.

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