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Vector concepts

Represent vectors in component form using i\mathbf{i} and j\mathbf{j}; find magnitude and direction; perform vector addition, subtraction and scalar multiplication; find unit vectors.

Worked examples

Magnitude, unit vector, direction angle

Straightforward

Problem

For v=(−4,3)\mathbf{v} = (-4, 3), find (a) the magnitude, (b) the unit vector, and (c) the angle v\mathbf{v} makes with the positive xx-axis.

Vector arithmetic

Moderate

Problem

Given a=3i−j\mathbf{a} = 3\mathbf{i} - \mathbf{j} and b=−i+4j\mathbf{b} = -\mathbf{i} + 4\mathbf{j}, find 2a−3b2\mathbf{a} - 3\mathbf{b} and its magnitude.

Magnitude-direction form to components

Challenging

Problem

A boat is travelling at 12 km/h on a bearing of 060° (measured clockwise from north). Express the velocity as a component vector in the form ai+bja\mathbf{i} + b\mathbf{j}, where i\mathbf{i} is east and j\mathbf{j} is north.

Practise

Q1·Straightforward
Find the magnitude of the vector v=3i+4j\mathbf{v} = 3\mathbf{i} + 4\mathbf{j}.
Q2·Straightforward
The vector u=6i+8j\mathbf{u} = 6\mathbf{i} + 8\mathbf{j}. Find the xx-component of the unit vector u^\hat{\mathbf{u}} in the direction of u\mathbf{u}.
Q3·Straightforward
Given v=3i−2j\mathbf{v} = 3\mathbf{i} - 2\mathbf{j}, find the vector 4v4\mathbf{v}. Express your answer as a coordinate pair (x,y)(x, y).
Q4·Straightforward
Vectors a=(2,5)\mathbf{a} = (2, 5) and b=(3,−1)\mathbf{b} = (3, -1). Find ∣a+b∣|\mathbf{a} + \mathbf{b}| to 2 decimal places.
Q5·Moderate
Given v=2i+3j\mathbf{v} = 2\mathbf{i} + 3\mathbf{j} and w=i−4j\mathbf{w} = \mathbf{i} - 4\mathbf{j}, find ∣3v−2w∣|3\mathbf{v} - 2\mathbf{w}| to 2 decimal places.
Q6·Moderate
A vector has magnitude 10 and makes an angle of 30° with the positive xx-axis. Find its xx-component to 2 decimal places.
Q7·Moderate
Find the angle (in degrees) that the vector v=−3 i+j\mathbf{v} = -\sqrt{3}\,\mathbf{i} + \mathbf{j} makes with the positive xx-axis. Give an exact answer.
Q8·Moderate
Given a=2i+5j\mathbf{a} = 2\mathbf{i} + 5\mathbf{j} and b=3i−j\mathbf{b} = 3\mathbf{i} - \mathbf{j}, find a+b\mathbf{a} + \mathbf{b}. Express your answer as a coordinate pair (x,y)(x, y).
Q9·Moderate
Three vectors satisfy a+b+c=0\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}. Given a=(2,−3)\mathbf{a} = (2, -3) and b=(−5,1)\mathbf{b} = (-5, 1), find ∣c∣|\mathbf{c}| to 4 decimal places.
Q10·Challenging
Two forces act on an object: F1=4i+6j\mathbf{F}_1 = 4\mathbf{i} + 6\mathbf{j} N and F2=3i−2j\mathbf{F}_2 = 3\mathbf{i} - 2\mathbf{j} N. Find the angle (to the nearest degree) that the resultant force makes with the positive xx-axis.
Q11·Challenging
A boat travels at 5 km/h in the direction of the vector 3i+4j3\mathbf{i} + 4\mathbf{j}. A current flows at 2 km/h in the direction −i-\mathbf{i}. Find the magnitude of the actual velocity of the boat to 4 decimal places.
Q12·Challenging
For v=2i+3j\mathbf{v} = 2\mathbf{i} + 3\mathbf{j}, find the positive value of λ\lambda such that ∣λv∣=10|\lambda\mathbf{v}| = 10. Give your answer to 4 decimal places.