Open MathNSWGitHub ↗

Dot product and geometry

Calculate the scalar (dot) product of two vectors; find the angle between vectors; test for perpendicularity and parallelism; find scalar and vector projections; prove geometric results using vectors.

Worked examples

Finding the angle between two vectors

Straightforward

Problem

Find the angle between u=(1,2)\mathbf{u} = (1, 2) and v=(3,−1)\mathbf{v} = (3, -1) to 2 decimal places.

Perpendicularity and finding an unknown

Moderate

Problem

Find the value of kk such that a=(k,3)\mathbf{a} = (k, 3) and b=(2,k−4)\mathbf{b} = (2, k - 4) are perpendicular.

Vector projection and geometric proof

Challenging

Problem

(a) Find the vector projection of a=(4,3)\mathbf{a} = (4, 3) onto b=(1,0)\mathbf{b} = (1, 0). (b) Prove that the diagonals of a rhombus are perpendicular.

Practise

Q1·Straightforward
Find the dot product a⋅b\mathbf{a} \cdot \mathbf{b} where a=(3,4)\mathbf{a} = (3, 4) and b=(2,1)\mathbf{b} = (2, 1).
Q2·Straightforward
Find a⋅b\mathbf{a} \cdot \mathbf{b} where a=(2,−3)\mathbf{a} = (2, -3) and b=(5,4)\mathbf{b} = (5, 4).
Q3·Straightforward
Find a⋅b\mathbf{a} \cdot \mathbf{b} where a=(3,−4)\mathbf{a} = (3, -4) and b=(4,3)\mathbf{b} = (4, 3).
Q4·Straightforward
Find the angle between a=(1,0)\mathbf{a} = (1, 0) and b=(1,1)\mathbf{b} = (1, 1). Give your answer in degrees.
Q5·Moderate
Find the angle between a=(3,4)\mathbf{a} = (3, 4) and b=(4,3)\mathbf{b} = (4, 3) to 2 decimal places. Give your answer in degrees.
Q6·Moderate
Find the positive value of kk such that the vectors (k, k2)(k,\, k^2) and (3, −1)(3,\, -1) are perpendicular.
Q7·Moderate
Find the scalar projection of a=(5,0)\mathbf{a} = (5, 0) onto b=(3,4)\mathbf{b} = (3, 4).
Q8·Moderate
Find the vector projection of a=(7,1)\mathbf{a} = (7, 1) onto b=(2,1)\mathbf{b} = (2, 1). Give your answer as a coordinate pair (x,y)(x, y).
Q9·Moderate
Find the value of kk such that the vectors (k+1, k)(k + 1,\, k) and (2, −3)(2,\, -3) are perpendicular.
Q10·Challenging
Points AA, BB, CC have position vectors a=(2,3)\mathbf{a} = (2, 3), b=(5,7)\mathbf{b} = (5, 7), c=(4,1)\mathbf{c} = (4, 1). Find the angle ∠ABC\angle ABC (at vertex BB) to 2 decimal places. Give your answer in degrees.
Q11·Challenging
A force F\mathbf{F} has magnitude 13 N and acts in the direction of the vector (5,12)(5, 12). An object is displaced by d=(4,1)\mathbf{d} = (4, 1) m. The work done is W=F⋅dW = \mathbf{F} \cdot \mathbf{d} (in joules). Find WW.
Q12·Challenging
Let a\mathbf{a} and b\mathbf{b} be vectors with ∣a∣=∣b∣|\mathbf{a}| = |\mathbf{b}|. Prove that (a+b)(\mathbf{a} + \mathbf{b}) and (a−b)(\mathbf{a} - \mathbf{b}) are perpendicular.

✎ Work this one through on paper — proofs are self-assessed.