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.

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

4√5

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

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