When two forces pull on the same object, or a plane flies through a crosswind, you need to combine vectors into a single resultant. And often you need the reverse — splitting one vector into two perpendicular pieces you can handle separately. This page covers both directions: adding vectors (triangle and parallelogram methods) and resolving them into components with trigonometry.
📚 What you need to know
Adding vectors gives a single resultant (or “net”) vector
The triangle method links vectors head-to-tail
The parallelogram method links vectors tail-to-tail
To subtract, reverse the vector’s direction and add
For perpendicular vectors, magnitude comes from Pythagoras, direction from trig
Resolving splits a vector into horizontal (Fcosθ) and vertical (Fsinθ) parts
Combining vectors: the triangle method
Two or more vectors can be added to produce a single resultant vector — the one vector that has the same overall effect. The most reliable way to picture this is the triangle method.
⚛ Triangle method
Link the vectors head-to-tail (the tail of the second starts at the head of the first).
Draw the resultant from the tail of the first vector to the head of the last.
To subtract, reverse the direction of the vector being taken away, then add as normal.
Link a and b head-to-tail; the resultant a + b runs from the start of the first vector to the end of the last.
The parallelogram method
An equivalent approach links the two vectors tail-to-tail (both starting from the same point), completes the parallelogram, and reads the resultant off the diagonal. It gives exactly the same answer — use whichever you find easier to draw.
Draw a and b from the same point, complete the parallelogram (dashed), and the resultant R is the diagonal.
Calculating a resultant (perpendicular vectors)
When the two vectors are at right angles, you don’t need a scale drawing at all — you can calculate the resultant exactly. The magnitude comes from Pythagoras and the direction from trigonometry.
Resultant of perpendicular vectorsR = √(a² + b²)θ = tan−1(b / a)
WE 1
Two forces act on a point at right angles: 5.0 N pointing east and 12.0 N pointing north. Find the magnitude and direction of the resultant.
Step 1 — magnitude (Pythagoras)R = √(5.0² + 12.0²) = √169 = 13.0 N
Step 2 — direction (trig)
θ = tan−1(12.0 / 5.0) = 67°13.0 N, 67° north of eastThe 5-12-13 combination is a classic right triangle — spotting it saves time, but the method works for any perpendicular pair.
Resolving vectors into components
Going the other way, a single vector can be resolved into two perpendicular components that, together, have the same effect. If a vector F makes an angle θ with the horizontal:
Components of a vectorhorizontal: Fx = F cosθvertical: Fy = F sinθ
The eternal question: is it sin or cos? Remember that cos pairs with the angle you’re measuring from — it gives the component lying alongside the angle (the adjacent side). Think “cos sandwich”: the flat horizontal layer of the sandwich is the cos component. Lock that in and sin is automatically the other one.
WE 2
A cable pulls a crate with a force of 60 N at 40° above the horizontal. Find the horizontal and vertical components of this force.
Step 1 — horizontal (cos)Fx = 60 × cos40° = 46 NStep 2 — vertical (sin)Fy = 60 × sin40° = 39 NFx = 46 N, Fy = 39 NCheck: √(46² + 39²) ≈ 60 N. The two components always rebuild the original vector.
Subtracting vectors
To subtract a vector, reverse its direction and then add it as normal. This matters most when finding a change in a vector — like the change in velocity when a ball bounces back off a wall.
Subtracting b is the same as adding −b: flip the arrow around, then combine head-to-tail as usual.
WE 3
A ball hits a wall horizontally at 8.0 m s−1 and rebounds straight back at 6.0 m s−1. Taking the initial direction as positive, find the change in velocity.
Step 1 — assign signs (direction matters)u = +8.0 m s−1, v = −6.0 m s−1 (reversed)
Step 2 — change = final − initial
Δv = v − u = −6.0 − (+8.0) = −14.0 m s−1Δv = 14 m s−1, opposite to the original motionThe rebound flips the sign, so the two speeds ADD in the subtraction. This is why bouncing gives a bigger velocity change than simply stopping.
💡 Top tips
Triangle method: link head-to-tail. Parallelogram: tail-to-tail.
Perpendicular vectors → use Pythagoras + trig, no drawing needed.
To subtract, reverse and add.
cos = component along the angle; sin = the other one.
Always give a resultant as magnitude AND direction.
⚠ Common mistakes
Adding vector magnitudes directly when they aren’t in the same direction
Mixing up sin and cos when resolving
Forgetting to reverse a vector before subtracting
Dropping the direction from the final answer
Using Pythagoras on vectors that aren’t perpendicular
Quick recap: Combine vectors with the triangle (head-to-tail) or parallelogram (tail-to-tail) method. For perpendicular vectors, magnitude = √(a²+b²) and direction = tan−1(b/a). Resolve a vector with Fx = Fcosθ and Fy = Fsinθ. To subtract, reverse and add.
Everything here works cleanly when vectors are perpendicular. But what if they meet at an awkward angle, like 50°? Then Pythagoras won’t do — you draw the vectors accurately to scale and measure the resultant with a ruler and protractor. That’s the technique in the next page: Scale Diagrams.
Vectors tying you in knots?
Book a free meeting and we’ll practise adding, subtracting and resolving until resultants and components feel automatic — the vector skills that show up all over Paper 1 and 2.
