Maths › Vectors › Vectors in two dimensions
Vectors in two dimensions
Some quantities need a direction as well as a size, and vectors carry both in one object. Arrows add tip to tail, components add slot by slot, and position vectors turn every geometry problem about points and midpoints into arithmetic you can do in the margin.
Builds on Straight lines and Triangles and the sine and cosine rules.
IN THIS TOPIC
- Work in column and i-j notation, converting between components and magnitude-direction form.
- Add vectors, scale them, and recognise parallel vectors.
- Use position vectors, AB = b − a, and distances to solve geometric problems.
WHAT YOU PROBABLY THINK
The magnitude of a + b is |a| + |b|.
Arrows with components
A vector has magnitude and direction both, and two notations carry it: the column form, components stacked, and the i-j form, multiples of the unit vectors i (one step across) and j (one step up). The magnitude |a| comes from Pythagoras on the components, and the direction from trigonometry, so a vector (3, 4) has magnitude 5 and points 53.1° above the horizontal. Dividing a vector by its own magnitude leaves a unit vector, length 1, same direction.
WORKED EXAMPLE
Between the two forms
Find the magnitude and direction of a = 3i + 4j, and the unit vector in the direction of a.
|a| = √(32 + 42) = 5.
Direction: tan θ = 4/3, so θ = 53.1° above the positive x-direction.
The unit vector is a divided by its length, (3/5)i + (4/5)j, whose own magnitude is 1 by construction.
Going the other way, a vector of magnitude 6 at 30° has components (6 cos 30°, 6 sin 30°) = (3√3, 3). The two forms are the same vector in different notation, and questions swap freely.
Adding, scaling, comparing
Vectors add tip to tail: walk along a, then along b, and the sum is the straight walk from start to finish. In components that is addition slot by slot. Multiplying by a scalar stretches a vector without turning it (or reverses it, if negative), so two vectors are parallel exactly when one is a scalar multiple of the other, 6i − 8j being 2(3i − 4j).
The picture also settles the opening lie. |a + b| is the length of the shortcut, and a shortcut beats the detour whenever the two legs point different ways: for a = (3, 4) and b = (4, −3), |a| + |b| = 10 while |a + b| = √50 ≈ 7.07. Magnitudes only add when the vectors already share a direction.
YOUR TURN
Parallel or not
Given p = 2i − 5j and q = −6i + 15j, show that p and q are parallel, and state the ratio of their magnitudes, before opening the working.
Show the working
q = −3p, a scalar multiple, so the two are parallel, pointing opposite ways because the scalar is negative.
|q| = 3|p|, so the ratio is 1 : 3.
Spotting the scalar is the whole method. Checking one component pair suggests it; checking both confirms it.
Position vectors and geometry
Fix an origin O and every point gets a position vector, a for the point A. The vector from A to B is then
destination minus start, and its magnitude is the distance between the points. This one identity turns geometry into component arithmetic: midpoints average position vectors, and shapes close when their side vectors match.
WORKED EXAMPLE
From one point to another
The points A(1, 5) and B(4, 1) have position vectors a and b. Find the vector AB and the distance AB.
AB = b − a = (4 − 1)i + (1 − 5)j = 3i − 4j.
The distance is |AB| = √(9 + 16) = 5.
Destination minus start, always. The reverse subtraction gives BA, the same walk reversed, and mixing them up flips every sign downstream.
TRY IT UNSEEN
Completing the parallelogram
ABCD is a parallelogram with A(2, 1), B(5, 3) and C(9, 2). Find the position vector of D.
Show the working
In a parallelogram AB = DC, so b − a = c − d.
Rearranging, d = a + c − b = (2 + 9 − 5)i + (1 + 2 − 3)j = 6i + 0j, the point (6, 0).
A check is built in: DC = c − d = 3i + 2j, which matches AB = 3i + 2j, so the shape genuinely closes.
Label order matters. ABCD names the corners in sequence around the shape, and a different order names a different parallelogram.
THE EXAM BIT
- Set out vector working in column or i-j form and stay in it; mixing notations mid-solution breeds sign errors.
- Magnitude is Pythagoras on components; direction is trigonometry, quoted against a stated reference direction.
- Parallel means scalar multiple; write the scalar down explicitly, sign included.
- AB = b − a, destination minus start; distances are magnitudes of such differences.
- In shape problems, translate the geometric fact into a vector equation (AB = DC for a parallelogram) before touching any numbers.
CHECK YOURSELF
Given a = 5i − 12j, find |a|, the unit vector in the direction of a, and a vector of magnitude 39 parallel to a.
Show a hint
5, 12 and a familiar hypotenuse; then scale.
Show the answer
|a| = √(25 + 144) = 13.
The unit vector is (5/13)i − (12/13)j.
Magnitude 39 is 3 × 13, so 3a = 15i − 36j works, as does −3a in the opposite direction, and saying so earns the final mark.
Components add slot by slot, scalars stretch, and parallel means scalar multiple.
AB = b − a, destination minus start, and its magnitude is the distance.
CHECK YOUR PROGRESS
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- Work in column and i-j notation, converting between components and magnitude-direction form.
- Add vectors, scale them, and recognise parallel vectors.
- Use position vectors, AB = b − a, and distances to solve geometric problems.
No animated video for this topic yet; these notes stand alone.