Vectors and Vector Arithmetic

Vectors describe movement

A vector has both size and direction. A translation vector has a horizontal component on top and a vertical component underneath. Right and up are positive; left and down are negative. A vector does not have a fixed starting point.

For example, $\left(\begin{array}{c}4\\ -2\end{array}\right)$ means move 4 units right and 2 units down. The same vector starting at P(−1,5) ends at Q(3,3).

Find a vector between two points

Subtract start coordinates from end coordinates. For A(2,−3) and B(−4,5), the horizontal change is −4−2=−6 and vertical change is 5−(−3)=8. Thus the vector from A to B is $\left(\begin{array}{c}-6\\ 8\end{array}\right)$.

Reversing the journey reverses both signs: the vector from B to A is $\left(\begin{array}{c}6\\ -8\end{array}\right)$. In symbols, BA=−AB, where bold pairs here denote directed vectors, not ordinary lengths.

Add and subtract vectors

Add matching components. Geometrically, place the second arrow at the end of the first: the sum is the single displacement from the initial start to the final end.

Worked example — two journeys

Let a=$\left(\begin{array}{c}3\\ 2\end{array}\right)$ and b=$\left(\begin{array}{c}-1\\ 4\end{array}\right)$. Then a+b=$\left(\begin{array}{c}2\\ 6\end{array}\right)$. Moving 3 right and 2 up, then 1 left and 4 up, gives 2 right and 6 up overall.

Worked example — subtraction

a−b means add the reverse of b. Subtract each component: (3−(−1), 2−4), giving $\left(\begin{array}{c}4\\ -2\end{array}\right)$. This is different from b−a, which is its negative.

Multiply by a scalar

A scalar is an ordinary number. Multiply both components by it. A positive scalar keeps the direction; a negative scalar reverses it. The magnitude is multiplied by the absolute value of the scalar.

Worked example

If u=$\left(\begin{array}{c}2\\ -3\end{array}\right)$, then 3u=$\left(\begin{array}{c}6\\ -9\end{array}\right)$ and −2u=$\left(\begin{array}{c}-4\\ 6\end{array}\right)$. Half of u is $\left(\begin{array}{c}1\\ -1.5\end{array}\right)$. Components need not be whole numbers.

The zero vector $\left(\begin{array}{c}0\\ 0\end{array}\right)$ represents no displacement. Adding it changes nothing.

Vector routes and missing movements

A journey from A to B then B to C is equivalent to going straight from A to C: AB+BC=AC. Directions must join head-to-tail.

Worked example

Given AB=$\left(\begin{array}{c}4\\ 1\end{array}\right)$ and AC=$\left(\begin{array}{c}7\\ 6\end{array}\right)$, find BC. Rearrange: BC=AC−AB=$\left(\begin{array}{c}3\\ 5\end{array}\right)$. Check by adding: (4,1)+(3,5)=(7,6).

A closed route has net vector zero because it returns to its starting point. For example AB+BC+CA=0, remembering that CA=−AC.

Magnitude and displacement

Magnitude means the length of an arrow. A vector $\left(\begin{array}{c}6\\ 8\end{array}\right)$ forms a right triangle with horizontal and vertical legs 6 and 8, so its magnitude is √(6²+8²)=10 units. The signs affect direction, not length.

Displacement is not always the total distance travelled. A journey 3 units east then 4 north has displacement $\left(\begin{array}{c}3\\ 4\end{array}\right)$ of magnitude 5, but its travelled distance is 7 units.

Think Like a Mathematician

Two arrows in different places can be equal vectors if their lengths and directions match. Position is irrelevant to equality of displacement.

Common Mistakes

Write horizontal change first. Subtract start from end. Reverse both signs for the opposite journey. Multiply both components by a scalar. Do not confuse a vector with its magnitude.

Practice — vector arithmetic

1. Translate (−2,4) by the column vector with components 5 and −3.

2. Find the vector from (3,−1) to (−2,6).

3. Add the vectors (2,−5) and (−7,3), written here as horizontal and vertical components.

4. Subtract (−2,4) from (5,1).

5. Multiply the vector (−3,2) by −2.

6. AB=(2,3), AC=(8,1). Find BC.

7. Find the magnitude of the vector (−5,12).

Frequently Asked Questions

Are vector calculations Higher-only?

No. Translations, column vectors, addition, subtraction and scalar multiplication are shared content. Geometric vector proofs are Higher.

Does a negative vector have negative length?

No. The minus sign reverses direction. Magnitude remains non-negative.

Continue Learning

Previous: Circle Theorems and Circle Proofs

Next: Vector Geometry and Proof

This completes the shared Foundation/Higher Geometry route. The next lesson develops Higher vector proofs.

Geometry and Measures topic overview