Understanding Vectors in Mathematics

When you picture a vector, think of an arrow drawn on a piece of paper – the length tells you how big it is, the tip shows you which way it points. In maths, that simple arrow packs two pieces of information: magnitude (the size) and direction (the orientation).

Formally we write a vector as \(\overrightarrow{AB}\), where A is the tail (where it starts) and B is the head (where it ends). The notation \(|\vec v|\) or \(||\vec v||\) stands for the magnitude, while the arrow above a symbol, like \(\vec F\), signals that the quantity has direction as well as size.

It helps to picture a real‑world scenario. Imagine a football coach directing a goalkeeper to pass the ball. The coach must tell the keeper two things: where to send the ball (the direction) and how hard to kick it (the magnitude). That combination – direction plus magnitude – is exactly what a vector represents.

In three‑dimensional space we break a vector down into components along the standard axes. Using the unit vectors \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\) we write
\(\displaystyle \vec A = x\hat{i} + y\hat{j} + z\hat{k}\)
where x, y and z are the magnitudes of the vector in the x‑, y‑ and z‑directions respectively. The tail of the vector sits at the origin (or any chosen point) and the head lands at the coordinate \((x, y, z)\).

In two dimensions the same idea applies, only we have just the x‑ and y‑components. If a vector \(\vec A\) makes an angle \(\theta\) with the positive x‑axis, then
\(A_x = A\cos\theta\) and \(A_y = A\sin\theta\).
The overall length follows from the Pythagorean theorem: \(|A| = \sqrt{A_x^2 + A_y^2}\).

Finding the angle between two vectors is a common task. The dot product links the magnitudes and the cosine of the angle:
\(\vec a \cdot \vec b = |\vec a|\,|\vec b|\cos\theta\).
Rearranging gives the familiar formula
\(\theta = \cos^{-1}\!\left(\frac{\vec a \cdot \vec b}{|\vec a|\,|\vec b|}\right)\).
So, by computing the dot product you can instantly discover how “tilted” one vector is relative to another.

Here are a few staple vector formulas you’ll keep reaching for:

• Addition / Subtraction: \((a\hat{i}+b\hat{j}+c\hat{k}) \pm (p\hat{i}+q\hat{j}+r\hat{k}) = (a\pm p)\hat{i} + (b\pm q)\hat{j} + (c\pm r)\hat{k}\).
• Dot product: \((a\hat{i}+b\hat{j}+c\hat{k})\cdot(p\hat{i}+q\hat{j}+r\hat{k}) = ap + bq + cr\).
• Cross product: \(\vec A \times \vec B = (br - cq)\hat{i} + (cp - ar)\hat{j} + (aq - bp)\hat{k}\).
• Angle: \(\theta = \cos^{-1}\!\left(\frac{\vec A \cdot \vec B}{|\vec A|\,|\vec B|}\right)\).
• Multiplication properties: dot product is commutative (\(\vec A\cdot\vec B = \vec B\cdot\vec A\)), but cross product is not – swapping the order flips the sign.

Some handy unit‑vector identities are worth memorising:
\(\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1\),
\(\hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{k}\cdot\hat{i}=0\),
\(\hat{i}\times\hat{j}=\hat{k}\), \(\hat{j}\times\hat{k}=\hat{i}\), \(\hat{k}\times\hat{i}=\hat{j}\).

Working through examples solidifies these ideas, so try breaking a vector into components, adding a couple of vectors, or finding the angle between two forces. Once you get the hang of it, vectors become a natural language for describing motion, fields, and even data in higher‑dimensional spaces.