MISSION · Geometry
Vectors
Quantities that need a direction as well as a size
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LEVEL 2 · LEARN
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The idea in one sentence
A vector carries both a size and a direction, so adding two of them means following one and then the other — not adding their numbers.
Vectors against scalars
A scalar is a plain number with a unit: 5 kg, 20 seconds, 30 °C.
A vector needs a direction too: a velocity of 30 mph north, a force of 10 N downwards, a displacement of 5 km east.
The distinction is practical. Walk 3 km east then 4 km north and you’ve travelled 7 km — that’s distance, a scalar. But you’re only 5 km from where you started, and that displacement is a vector. The two answers are both correct and answer different questions.
Writing them down
A vector in the plane is usually written as components — how far across, how far up:
\[\mathbf{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\]Its magnitude is its length, straight from Pythagoras:
\[|\mathbf{a}| = \sqrt{3^2 + 4^2} = 5\]Position on the page doesn’t matter. Two arrows of the same length pointing the same way are the same vector, wherever they’re drawn — which is what makes the tip-to-tail method below legitimate.
Adding: tip to tail
Draw the first vector, start the second where the first ended, and the resultant runs from the original start to the final finish.
In components it’s simply across-plus-across and up-plus-up:
\[\begin{pmatrix} 3 \\ 4 \end{pmatrix} + \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}\]Note what the magnitudes do: 5 and \(\sqrt{5} \approx 2.24\) combine into \(\sqrt{34} \approx 5.83\), not 7.24. Magnitudes don’t add unless the vectors point the same way. Two forces of 10 N can total anything from 0 to 20 N depending on their directions, and that’s the whole reason vectors exist as a concept.
Multiplying by a scalar
\(2\mathbf{a}\) is twice as long, same direction. \(-\mathbf{a}\) is the same length, opposite direction. So subtraction is addition of the reverse: \(\mathbf{a} - \mathbf{b}\) means follow \(\mathbf{a}\), then go backwards along \(\mathbf{b}\).
If two vectors are scalar multiples of each other, they’re parallel — which is how most exam questions about vectors and geometry get proved.
Where they’re used
Anything with a direction: forces in a bridge, wind against an aircraft’s heading, a character’s motion in a game engine, the gradient of a surface. Whenever a quantity can partly cancel another out, it’s a vector, and adding the sizes would give the wrong answer.
Common mix-ups
- Adding magnitudes. Add components, then find the magnitude of the result.
- Confusing distance with displacement. A closed loop has distance but zero displacement.
- Ignoring the sign of a component. Down and left are negative.
- Thinking position matters. Same length, same direction, same vector.
Check yourself
Add \(\begin{pmatrix} 4 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ 5 \end{pmatrix}\), and find the magnitude of the result.
Components add to \(\begin{pmatrix} 3 \\ 6 \end{pmatrix}\), whose magnitude is \(\sqrt{9 + 36} = \sqrt{45} \approx 6.71\) — less than the two magnitudes summed.
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