In this page, you will learn about the bilinear property of vector dot product.

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Dot product is defined between `vec p, vec q in bbb V` and the result `vec p cdot vec q in RR`. This can be considered as a transformation. What type of transformation is this?

- unary
- binary
- binary
- ternary

The answer is 'binary', as the input is two vectors. 'bi' means two.

Which of the following property is true for Bilinear transformation?

- `T(x+y, z)=T(x,z)+T(y,z)`
- `T(ax,z)=aT(x,z)`
- `T(ax+y, z)= aT(x,z)+T(y,z)`
- All the above
- All the above

The answer is 'All the above'.

This property defines linearity for binary operator `T`. That is why it is named Bilinear.

Consider the dot product as a binary transform `T(vec p, vec q) = vec p cdot vec q`. Is dot product bilinear?

- Yes
- Yes
- No

The answer is 'Yes'. The dot product is distributive over vector addition and also satisfies the scalar multiple property.

• Dot product is bilinear.

**Bilinear Property: ** For any vector `vec p, vec q, vec r in bbb V` and `lambda in RR`

`(lambda vec p + vec q) cdot vec r = lambda (vec p cdot vec r) + (vec q cdot vec r)`

*Solved Exercise Problem: *

Given `vec x cdot vec z = 2` and `vec y cdot vec z = 1`, what is `(3 vec x + 2 vec y) cdot vec z`?

- `5`
- `6`
- `8`
- `8`
- `9`

The answer is '`8`'

`(3 vec x + 2 vec y) cdot vec z`

`quad = 3 vec x cdot vec z + 2 vec y cdot vec z`

*slide-show version coming soon*