# Vector Addition

To add two vectors, add their coordinates or components.

#### Examples

Given = (2, 1, 3), = (1, −1, 0) and = (1, 2, 3), find the vector = 2u + 3v − w.

= (4, 2, 6) + (3, −3, 0) − (1, 2, 3) = (6, −3, 3)

Given vectors and , determine the magnitude of the vector .

#### Properties of Vector Addition

Associative.

+ ( + ) = ( + ) +

Commutative.

+ = +

Additive identity.

+ =

Additive inverse or opposite.

+ (− ) =

### Scalar Multiplication

The product of a number k by a vector is another vector:

In the same direction as if k is positive.

In opposite direction as if k is negative.

Of magintude .

#### Properties

Associative.

**k · (k' · ) = (k · k') · **

Right distributivity.

**k · ( + ) = k · + k · **

Left distributivity.

**(k + k') · = k · + k' · **

Multiplicative Identity.

**1 · = **

#### Example

Given = (6, 2, 0), determine so that 3 = .