Magnitude and Direction of a Vector · Mathematics

Displacement and vectors

A vector has magnitude (size) and direction. A displacement is a vector. A scalar has magnitude only. A vector can be written as a column vector (x over y) or as xi + yj, where i and j are unit vectors along the axes.

The magnitude of a = (x, y) and the angle θ it makes with the x-axis.

Operations on vectors

Operation	Rule	Example
Addition	Add the components	(2, 3) + (1, −4) = (3, −1)
Subtraction	Subtract the components	(2, 3) − (1, −4) = (1, 7)
Scalar multiple	Multiply each component	3(2, 3) = (6, 9)
Position vector	AB = b − a	A(1, 2), B(4, 6): AB = (3, 4)

Worked exampleMagnitude and direction

Find the magnitude and direction of a = (3, 4).

Magnitude.
Direction.

Answer: Magnitude 5, direction 53.1° above the x-axis.

Vectors help to prove geometrical facts. The midpoint of AB has position vector ½(a + b). Two vectors are parallel if one is a scalar multiple of the other. Three points are collinear if AB is a multiple of AC.

Try itSum of two vectors

Choose the components of two vectors.

Check yourself The magnitude of (6, 8) is

Check yourself (2, 5) + (3, −1) =

Check yourself If A(1, 2) and B(5, 8), then AB =

Key terms

Vector: A quantity with size and direction.
Scalar: A quantity with size only.
Position vector: The vector from the origin to a point.
Resultant: The sum of two or more vectors.

Summary

Vectors have magnitude and direction; scalars have only magnitude.
Add and subtract by components.
|a| = √(x² + y²); direction from tan θ = y/x.
AB = b − a; parallel vectors are scalar multiples.