 # Vectors for SQA National 5 Maths

1. Vector magnitude

The magnitude of a vector can be calculated using Pythagoras’ theorem. A vector given by (x y) has magnitude √(x² + y²), and a vector given by (x y z) has magnitude √(x² + y² + z²). To obtain the distance between two points, the vector between them can be found and then its magnitude can be calculated to give the required distance. # 1

Find the distance between the two plotted points.

AB = (12 − 4 5 − 9) = (8 −4)
The distance can be calculated using Pythagoras’ theorem.
distance = √(8² + (−4)²) = 4√5

4√5 # 2

Find the value of x, given that the distance between (1, 3, 7) and (x, 2, 1) is √53.

Let (1, 3, 7) be A and (x, 2, 1) be B.
AB = (x − 1 2 − 3 1 − 7) = ((x − 1) −1 −6)

The distance can be calculated using Pythagoras’ theorem.
√53 = √((x − 1)² + (−1)² + (−6)²)
53 = (x − 1)² + 1 + 36
(x − 1)² = 16
x − 1 = 4
x = 4 + 1 = 5

5 # 3

Nina has drawn points (1, 4) and (4, 7) on a coordinate axis. Find the distance between the two points.

vector between the points = (4 − 1 7 − 4) = (3 3)
The distance can be calculated using Pythagoras’ theorem.
distance = √(3² + (−3)²) = 3√2

3√2 End of page