3D Vectors

3D Vectors behave in the same way as 2D Vectors, except that there is a third dimension to take into account.

For a column vertex, the `z`-direction is added as a third row eg `vec(AB) = ((3),(5),(7))`, where the 7 is the movement in the `z`-direction.

The magnitude of the vector is given by Pythagoras` Theorem applied in three dimension.

3D Vector

Example 1

Point A is a point on a three dimension graph at position (1, 2 ,3). Point B is a point on the same graph at position (4, 5, 6). Show `vec(AB)` as a column vector.

The `x` value is 4 - 1 = 3

Similarly, `y` is 5 - 2 = 3 and `z` is 6 - 3

Answer: `((3),(3),(3))`

Example 2

Vector `vec(AB) = ((3),(4),(5))`. What is the magnitude of `vec(AB)`? Give your answer to 2 decimal places.

Use Pythagoras` Theorem in 3D.

Pythagoras	`a^2`	`= b^2 + c^2 + d^2`
substitute	`a^2`	`= 3^2 + 3^2 + 3^2`
	`a`	`= sqrt(27)`
	`a`	`= 5.196`
2 dp	`a`	`= 5.20`

Answer: 5.20