🧭 Navigating Vectors: Mastering Position & Scalar Multiplication!

Vectors are the language of direction and magnitude, allowing us to map journeys, describe forces, and solve geometric puzzles. **Understanding vectors equips you to tackle coordinate geometry, mechanics, and even physics questions in your O Level exams.**

Key Concepts: Position Vectors & Scalar Multiplication

A **position vector** locates a point

P(x, y)

from the origin

O

\vec{OP} = \begin{pmatrix}x\\y\end{pmatrix}

. **Scalar multiplication** stretches or shrinks a vector: for any scalar

k

k\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}kx\\ky\end{pmatrix}.

These ideas combine to solve many geometric problems.

Adding vectors: $\vec{a} + \vec{b}$ gives a resultant.
Subtracting vectors: $\vec{a} - \vec{b}$ represents moving from $\vec{b}$ to $\vec{a}$ .

Worked Example: Triangle Navigation

Problem: In

\triangle ABC

\vec{OA}=\begin{pmatrix}4\\1\end{pmatrix}

and

\vec{OB}=\begin{pmatrix}1\\5\end{pmatrix}

. Point

D

is the midpoint of

AB

. Find (i)

\vec{AB}

, (ii)

\vec{OD}

, and (iii)

\vec{CD}

C

is such that

\vec{OC}=2\vec{OD}

Step 1:
$\vec{AB}=\vec{OB}-\vec{OA}=\begin{pmatrix}1\\5\end{pmatrix}-\begin{pmatrix}4\\1\end{pmatrix}=\begin{pmatrix}-3\\4\end{pmatrix}.$
Step 2:
$\vec{OD}=\vec{OA}+\frac{1}{2}\vec{AB}=\begin{pmatrix}4\\1\end{pmatrix}+\frac{1}{2}\begin{pmatrix}-3\\4\end{pmatrix}=\begin{pmatrix}4-1.5\\1+2\end{pmatrix}=\begin{pmatrix}2.5\\3\end{pmatrix}.$
Step 3:
$\vec{OC}=2\vec{OD}=2\begin{pmatrix}2.5\\3\end{pmatrix}=\begin{pmatrix}5\\6\end{pmatrix},$
so
$\vec{CD}=\vec{OD}-\vec{OC}=\begin{pmatrix}2.5\\3\end{pmatrix}-\begin{pmatrix}5\\6\end{pmatrix}=\begin{pmatrix}-2.5\\-3\end{pmatrix}.$

Now that you have seen vectors in action, **practise plotting points and performing operations until the processes become second nature.** The more you visualise and calculate, the faster you will master vector problems in your examinations!