Learning Objectives
  • Use column vectors and directed line-segment notation.
  • Add and subtract vectors and multiply a vector by a scalar.
  • Calculate the magnitude of a two-dimensional vector.
  • Use position vectors and express unknown vectors in terms of given vectors.
  • Prove parallelism or collinearity and solve ratio problems using vectors.
Key Terms
Vector
A quantity with magnitude and direction.
Scalar
A quantity with magnitude only.
Column vector
A vector written as vertical horizontal and vertical components.
Position vector
The vector from the origin to a point.
Magnitude
The length of a vector.
Collinear
Lying on the same straight line.
Parallel vectors
Vectors that are scalar multiples of one another.
Resultant
The single vector equivalent to a combination of vectors.
What This Chapter Covers
  • Column-vector arithmetic
  • Magnitude and direction
  • Position vectors and route methods
  • Ratios on line segments
  • Geometric proof of parallelism and collinearity
Vectors And Scalars

A vector records both size and direction. A displacement of 5 km east is a vector; a distance of 5 km is a scalar. In two dimensions, a column vector \begin{pmatrix}x\\y\end{pmatrix} means x units horizontally and y units vertically. Positive x is right and positive y is up.

\mathbf{a}=\begin{pmatrix}x\\y\end{pmatrix}
Adding And Subtracting Vectors

Add corresponding components. Geometrically, vector addition follows the triangle or parallelogram rule. Subtracting a vector is equivalent to adding the vector in the opposite direction.

\begin{pmatrix}a\\b\end{pmatrix}+\begin{pmatrix}c\\d\end{pmatrix}=\begin{pmatrix}a+c\\b+d\end{pmatrix}
\begin{pmatrix}a\\b\end{pmatrix}-\begin{pmatrix}c\\d\end{pmatrix}=\begin{pmatrix}a-c\\b-d\end{pmatrix}
Worked Example: Component Arithmetic

Question: Given \mathbf{p}=\begin{pmatrix}4\\-3\end{pmatrix} and \mathbf{q}=\begin{pmatrix}-2\\5\end{pmatrix}, find 2\mathbf{p}-\mathbf{q}.

  1. 2\mathbf{p}=\begin{pmatrix}8\\-6\end{pmatrix}.
  2. Subtract q componentwise.
  3. \begin{pmatrix}8-(-2)\\-6-5\end{pmatrix}=\begin{pmatrix}10\\-11\end{pmatrix}.

Answer: \begin{pmatrix}10\\-11\end{pmatrix}.

Scalar Multiplication

Multiplying by a positive scalar changes the magnitude but keeps the direction. Multiplying by a negative scalar reverses the direction. If \mathbf{b}=k\mathbf{a}, then a and b are parallel; if k is positive they point in the same direction, and if k is negative they point in opposite directions.

Worked Example: Identify Parallel Vectors

Question: Show that \mathbf{a}=\begin{pmatrix}3\\-2\end{pmatrix} and \mathbf{b}=\begin{pmatrix}-12\\8\end{pmatrix} are parallel.

  1. \mathbf{b}=-4\begin{pmatrix}3\\-2\end{pmatrix}=-4\mathbf{a}.
  2. Because one vector is a scalar multiple of the other, they are parallel.
  3. The negative scalar means they point in opposite directions.

Answer: They are parallel in opposite directions.

Magnitude Of A Vector

The components form the perpendicular sides of a right triangle. Use Pythagoras’ theorem.

\left|\begin{pmatrix}x\\y\end{pmatrix}\right|=\sqrt{x^2+y^2}
Worked Example: Magnitude

Question: Find the magnitude of \begin{pmatrix}-5\\12\end{pmatrix}.

  1. Square the components: (-5)^2=25 and 12^2=144.
  2. Add: 25+144=169.
  3. Take the square root.

Answer: 13.

Route Method

For points A, B and C, \overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}. Reversing a vector changes its sign: \overrightarrow{BA}=-\overrightarrow{AB}. Choose a route whose component vectors are known.

Worked Example: Express A Vector

Question: Given \overrightarrow{OA}=\mathbf{a} and \overrightarrow{OB}=\mathbf{b}, express \overrightarrow{AB}.

  1. Use the route A to O to B.
  2. \overrightarrow{AO}=-\mathbf{a}.
  3. \overrightarrow{OB}=\mathbf{b}.
  4. Add the route vectors.

Answer: \overrightarrow{AB}=\mathbf{b}-\mathbf{a}.

Division Of A Line Segment

If point P divides AB in the ratio AP:PB=m:n, then P is \frac{m}{m+n} of the way from A to B. Using position vectors \mathbf{a} and \mathbf{b}, the position vector of P is a weighted average.

\overrightarrow{OP}=\frac{n\mathbf{a}+m\mathbf{b}}{m+n}\quad\text{when }AP:PB=m:n
Worked Example: Internal Division

Question: A and B have position vectors a and b. Point P divides AB in the ratio AP:PB=2:3. Find \overrightarrow{OP}.

  1. P is \frac25 of the way from A to B.
  2. \overrightarrow{OP}=\mathbf{a}+\frac25(\mathbf{b}-\mathbf{a}).
  3. Simplify.

Answer: \overrightarrow{OP}=\frac35\mathbf{a}+\frac25\mathbf{b}=\frac{3\mathbf{a}+2\mathbf{b}}5.

Proving Collinearity

To prove A, B and C are collinear, show that \overrightarrow{AB}=k\overrightarrow{AC} for some scalar k. The vectors must share a common starting point. If 0\lt k\lt1, B lies between A and C.

Worked Example: Collinearity

Question: Suppose \overrightarrow{OP}=2\mathbf{a}+\mathbf{b}, \overrightarrow{OQ}=5\mathbf{a}+4\mathbf{b} and \overrightarrow{OR}=8\mathbf{a}+7\mathbf{b}. Show P, Q and R are collinear.

  1. \overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}=3\mathbf{a}+3\mathbf{b}=3(\mathbf{a}+\mathbf{b}).
  2. \overrightarrow{QR}=\overrightarrow{OR}-\overrightarrow{OQ}=3\mathbf{a}+3\mathbf{b}=3(\mathbf{a}+\mathbf{b}).
  3. The vectors PQ and QR are equal and parallel, so P, Q and R lie on one straight line.

Answer: P, Q and R are collinear, with Q the midpoint of PR.

Vector Proof In Geometric Figures

Use two independent base vectors, express every required route in terms of them, and simplify. A proof is complete only when the vector relationship is interpreted geometrically, such as “therefore the lines are parallel” or “therefore the points are collinear”.

Examination Guidance
  • Keep arrow direction consistent: \overrightarrow{AB}=-\overrightarrow{BA}.
  • When proving parallel lines, state the scalar-multiple relationship explicitly.
  • For collinearity, compare vectors with the same starting point or along the same line.
  • Use exact fractions in ratio problems.
  • End a vector proof with a sentence explaining the geometric conclusion.
Common Mistakes
  • Adding position vectors directly when the required vector is between two points; use endpoint minus start point.
  • Using Pythagoras on vector components without squaring negative values correctly.
  • Saying vectors are equal when they are merely parallel scalar multiples.
  • Reversing a ratio when forming a weighted average.
  • Omitting the final geometric interpretation in a proof.
Knowledge Check And Practice

1. Find \begin{pmatrix}3\\4\end{pmatrix}+\begin{pmatrix}-1\\2\end{pmatrix}.

Answer: \begin{pmatrix}2\\6\end{pmatrix}.

2. Find the magnitude of \begin{pmatrix}8\\15\end{pmatrix}.

Answer: 17.

3. If \overrightarrow{OA}=\mathbf{a} and \overrightarrow{OB}=\mathbf{b}, express \overrightarrow{BA}.

Answer: \mathbf{a}-\mathbf{b}.

4. What relationship proves two vectors are parallel?

Answer: One is a scalar multiple of the other.

5. P divides AB in the ratio 1:3. Express OP in terms of a and b.

Answer: \frac34\mathbf{a}+\frac14\mathbf{b}.

6. If \overrightarrow{AB}=2\overrightarrow{AC}, what can be concluded about A, B and C?

Answer: They are collinear; the exact ordering depends on the sign and magnitude of the scalar.
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