__maths__>__Properties of Vector Arithmetics__>__Properties of Vector Multiplication by Scalar__### Distributive over Vector Addition

In this page, the property, multiplication of scalar is distributive over vector addition, is explained.

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Considering that a vector has components as real numbers, Given that `vec p = ai+bj+ck`, `vec q = e i + fj + g k` and `lambda ` where `a,b,c,d,e,f,lambda in bbb R`, What will be the result of `lambda (vec p + vec q)`?

- scalar
- `lambda vec p + lambda vec q`
- `lambda vec p + lambda vec q`
- `lambda vec p + vec q`

The answer is '`lambda vec p + lambda vec q`'. This is proven by multiplying the components individually by the scalar and using vector addition properties.

• A scalar multiplied by sum of two vectors equals sum of vectors multiplied by the scalar.

**Distributive over Vector Addition: ** Given scalar `lambda` and vectors `vec p, vec q`,

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

*Solved Exercise Problem: *

Given that `vec p = 6 vec q` and `vec r = 4 vec s`, which of the following equals `3 ( 2vec q+ 4/3 vec s)`?

- `vec p + vec r`
- `vec p + vec r`
- `vec p + 2 vec r`
- `vec p - vec r`
- `vec r - vec p`

The answer is '`vec p + vec r`'.

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