Vectors in 2D - Addition, Subtraction, Scalar Multiplication

A vector carries two pieces of information at once — a magnitude and a direction — and this chapter builds the algebra that lets you combine vectors exactly as reliably as you combine numbers. Where the previous chapter described points and lines through coordinates, this chapter introduces a new object, the vector, that describes displacement, and the operations that let you add, subtract and scale displacements.

To follow the chapter you need the distance formula and basic coordinate geometry from the previous chapter, since a vector's magnitude is computed exactly the same way. Have graph paper or a sketch ready — vectors are easiest to understand by drawing them before computing with them.

A vector is not a point. A point $(3,4)$ names a fixed location. A vector $\begin{pmatrix}3\\4\end{pmatrix}$ names a displacement — "3 units across, 4 units up" — and the same vector can be drawn starting from any point in the plane; only its length and direction matter, not where it is drawn.

1. Vectors as Displacements

A vector is a quantity possessing both magnitude and direction, represented geometrically as an arrow and written $\vec{v}$ or $\mathbf{v}$. The vector from point $A$ to point $B$ is written $\overrightarrow{AB}$, and if $A(x_1,y_1)$ and $B(x_2,y_2)$, its component form is

— exactly the differences used inside the distance formula. Two vectors are equal if and only if they have the same components, regardless of where they are drawn; $\overrightarrow{AB}$ and $\overrightarrow{CD}$ represent the same vector whenever $B-A = D-C$, even if $A\neq C$.

A position vector is the special case of a vector drawn from the origin $O$ to a point $P(x,y)$, written $\overrightarrow{OP}$ or $\vec{p}$, with components simply $\begin{pmatrix}x\\y\end{pmatrix}$. Position vectors let every point be treated as a vector, which is what makes vector algebra useful for geometry.

2. Magnitude of a Vector

The magnitude (or length, or norm) of $\vec{v} = \begin{pmatrix}a\\b\end{pmatrix}$ is denoted $|\vec{v}|$ and found by Pythagoras' theorem, treating $a$ and $b$ as the legs of a right triangle:

A vector with $|\vec{v}|=1$ is called a unit vector. The magnitude of $\overrightarrow{AB}$ is exactly the distance $|AB|$ from the previous chapter — vectors and coordinate geometry describe the same underlying facts in different language.

Worked example. Find $|\vec{v}|$ for $\vec{v} = \begin{pmatrix}-5\\12\end{pmatrix}$.

3. Addition and Subtraction of Vectors

Vectors add component-wise: to add $\vec{u}=\begin{pmatrix}u_1\\u_2\end{pmatrix}$ and $\vec{v}=\begin{pmatrix}v_1\\v_2\end{pmatrix}$,

Geometrically this is the triangle rule: draw $\vec{u}$, then draw $\vec{v}$ starting where $\vec{u}$ ended; the sum is the vector from the start of $\vec{u}$ to the end of $\vec{v}$. Equivalently, the parallelogram rule draws both vectors from the same starting point and completes the parallelogram; the diagonal from that common point is the sum. Both rules describe the same addition, since in both pictures the resulting arrow has components $u_1+v_1$ and $u_2+v_2$.

Subtraction $\vec{u}-\vec{v}$ is addition of the negative of $\vec{v}$, where $-\vec{v}=\begin{pmatrix}-v_1\\-v_2\end{pmatrix}$ is the vector of the same magnitude pointing in the opposite direction:

Geometrically, if $\vec{u}$ and $\vec{v}$ are drawn from the same point, $\vec{u}-\vec{v}$ is the vector from the tip of $\vec{v}$ to the tip of $\vec{u}$ — the "other" diagonal of the parallelogram. This is exactly why $\overrightarrow{AB} = \vec{b}-\vec{a}$ for position vectors $\vec{a}=\overrightarrow{OA}$ and $\vec{b}=\overrightarrow{OB}$: travelling from $A$ to $B$ is the same as travelling from $O$ to $B$ and subtracting the journey from $O$ to $A$.

Worked example. Let $\vec{u}=\begin{pmatrix}4\\-1\end{pmatrix}$ and $\vec{v}=\begin{pmatrix}-2\\3\end{pmatrix}$. Find $\vec{u}+\vec{v}$ and $\vec{u}-\vec{v}$.

4. Scalar Multiplication

Multiplying a vector $\vec{v}=\begin{pmatrix}v_1\\v_2\end{pmatrix}$ by a real number (scalar) $k$ scales every component:

If $k>0$, $k\vec{v}$ points in the same direction as $\vec{v}$ but is $k$ times as long. If $k<0$, $k\vec{v}$ points in the exact opposite direction. If $k=0$, the result is the zero vector $\vec{0}=\begin{pmatrix}0\\0\end{pmatrix}$, which has no direction at all.

Parallel vectors. Two non-zero vectors $\vec{u}$ and $\vec{v}$ are parallel if and only if $\vec{u}=k\vec{v}$ for some scalar $k$ — equivalently, their components are proportional: $\dfrac{u_1}{v_1}=\dfrac{u_2}{v_2}$ (provided neither denominator is zero). This is the vector analogue of two lines sharing a gradient.

Worked example — the harder case. Determine whether $\vec{u}=\begin{pmatrix}6\\-9\end{pmatrix}$ is parallel to $\vec{v}=\begin{pmatrix}-4\\6\end{pmatrix}$, and if so, find $k$ such that $\vec{u}=k\vec{v}$.

Checking proportionality: $\dfrac{6}{-4}=-\dfrac{3}{2}$ and $\dfrac{-9}{6}=-\dfrac{3}{2}$. The ratios match, so the vectors are parallel, with $k=-\dfrac{3}{2}$ (confirm: $-\dfrac32\begin{pmatrix}-4\\6\end{pmatrix}=\begin{pmatrix}6\\-9\end{pmatrix}$ ✓). The edge case to notice: the scalar is negative, so despite being "parallel," $\vec{u}$ and $\vec{v}$ actually point in opposite directions — "parallel" in the vector sense includes anti-parallel vectors, unlike the everyday use of the word.

5. Unit Vectors and the $\vec{i}$, $\vec{j}$ Notation

The vectors $\vec{i}=\begin{pmatrix}1\\0\end{pmatrix}$ and $\vec{j}=\begin{pmatrix}0\\1\end{pmatrix}$ are the standard unit vectors along the two axes. Any vector can be written as a combination of them:

This is a linear combination, and it is exactly how component form and $\vec{i}$,$\vec{j}$ form express the same vector. To find the unit vector $\hat{v}$ in the direction of a non-zero vector $\vec{v}$, divide by its own magnitude:

Worked example. Find the unit vector in the direction of $\vec{v}=\begin{pmatrix}3\\4\end{pmatrix}$.

Since $|\vec{v}|=\sqrt{9+16}=5$,

Check: $|\hat{v}|=\sqrt{0.36+0.64}=\sqrt{1}=1$ ✓.

6. Position Vectors and the Midpoint

If $\vec{a}$ and $\vec{b}$ are the position vectors of $A$ and $B$, the position vector of the midpoint $M$ of $AB$ is

This restates the midpoint formula of the previous chapter in vector language, and it generalises immediately: the point $P$ dividing $AB$ in the ratio $m:n$ has position vector

identical in structure to the coordinate section formula, because each component of a vector equation is itself just that coordinate equation.

Common Exam Pitfalls

• Treating a vector like a point. A vector has no fixed location; writing $\vec{v}=(3,4)$ as if it were the point $(3,4)$ blurs a distinction the exam rewards you for keeping — use column notation or $\overrightarrow{AB}$ to signal a vector clearly.
• Computing $\overrightarrow{AB}$ as $A-B$ instead of $B-A$. The vector from $A$ to $B$ is "where $B$ is relative to $A$," so it is always the destination's position vector minus the start's: $\vec{b}-\vec{a}$.
• Forgetting that scalar multiplication by a negative number reverses direction. $-2\vec{v}$ is not merely "longer" — it points the opposite way, which matters when sketching or interpreting a result geometrically.
• Confusing "parallel" with "same direction." Vectors satisfying $\vec{u}=k\vec{v}$ with $k<0$ are still classified as parallel in this algebraic sense, even though they point in opposite directions.
• Dividing by magnitude instead of multiplying by its reciprocal when finding a unit vector — and forgetting to check the result has length 1. Always verify $|\hat v| = 1$ as a final check.
• Adding magnitudes instead of adding vectors. $|\vec{u}+\vec{v}| \neq |\vec{u}|+|\vec{v}|$ in general — only when $\vec u$ and $\vec v$ point in exactly the same direction does equality hold; otherwise the triangle inequality makes the sum's magnitude strictly smaller.

Summary

• A vector $\overrightarrow{AB}$ with $A(x_1,y_1)$, $B(x_2,y_2)$ has components $\begin{pmatrix}x_2-x_1\\y_2-y_1\end{pmatrix}$, and $\overrightarrow{AB}=\vec b-\vec a$ in position-vector form.
• Magnitude: $|\vec v|=\sqrt{a^2+b^2}$ for $\vec v=\begin{pmatrix}a\\b\end{pmatrix}$; this equals the distance formula.
• Addition/subtraction is component-wise; geometrically, the triangle rule (addition) and parallelogram rule (both diagonals) describe the same algebra.
• Scalar multiplication $k\vec v$ scales magnitude by $|k|$ and reverses direction when $k<0$.
• Two non-zero vectors are parallel iff $\vec u = k\vec v$ for some scalar $k$ (proportional components); this includes anti-parallel vectors ($k<0$).
• Unit vectors: $\vec i=\begin{pmatrix}1\\0\end{pmatrix}$, $\vec j=\begin{pmatrix}0\\1\end{pmatrix}$; any vector is $a\vec i+b\vec j$; the unit vector in the direction of $\vec v$ is $\hat v = \vec v/|\vec v|$.
• Midpoint in vector form: $\vec m=\tfrac12(\vec a+\vec b)$; section-ratio form: $\vec p = \dfrac{n\vec a+m\vec b}{m+n}$.