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Vectors

14 min read

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.

Adding vectorsTwo vectors drawn head to tail. Vector A is 4 units to the right and vector B is 3 units upwards. The resultant is the straight line from the start of A to the end of B, which is 5 units long. A parallelogram gives the same result.A = 4B = 3resultant RR = √(A² + B²) = √(16 + 9) = 5parallelogram method
The magnitude of a = (x, y) and the angle θ it makes with the x-axis.

Operations on vectors

OperationRuleExample
AdditionAdd the components(2, 3) + (1, −4) = (3, −1)
SubtractionSubtract the components(2, 3) − (1, −4) = (1, 7)
Scalar multipleMultiply each component3(2, 3) = (6, 9)
Position vectorAB = b − aA(1, 2), B(4, 6): AB = (3, 4)
Resolving a vector into componentsA force of 10 newtons acts at 30 degrees above the horizontal. It can be replaced by a horizontal component of 10 cosine 30, which is 8.7 newtons, and a vertical component of 10 sine 30, which is 5 newtons. The two components together have the same effect as the original force.θ = 30°F = 10 NF cos θ = 8.7 NF sin θ = 5 N
Worked exampleMagnitude and direction

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

  1. Magnitude.
  2. 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.
Trigonometry Matrices and Transformations