In this page, learn about adding direction to the scalar product with an example.

*click on the content to continue..*

An object by position vector `vec p` casts a shadow on a screen. The screen in given by unit vector `vec s` Is the shadow a vector or scalar?

- a vector
- a vector
- a scalar

The answer is 'a vector'. The shadow is a vector that spans as a ray with a starting and end point.

`vec p` and the screen `hat s` are given. what is the length of the shadow?

- `vec p cdot hat s`
- length is a scalar `|p| cos theta`
- both the above
- both the above

The answer is 'Both the above'.

An object by position vector `vec p` casts a shadow on a screen. The screen in given by unit vector `vec s`. We saw that the shadow is a vector and the length of the shadow is `vec p cdot hat s`. Is the shadow as a vector?

- `(vec p cdot hat s) hat s`
- `(vec p cdot hat s) hat s`
- `(vec p cdot hat s) hat p`
- `(vec p cdot hat s) vec p`

The answer is '`(vec p cdot hat s) hat s`'.

Vector dot product by definition is a scalar. Depending on the application requirement, a direction can be added to the scalar.

** Projection vector using Dot Product: ** For a vector `vec p` and a direction given by unit vector `hat s`, the projection vector of `vec p` in direction `hat s` is `(vec p cdot hat s) hat s`.

*slide-show version coming soon*