Scalar and Vector Quantities
1. Vector language and arithmetic
Vectors encode displacement: direction and magnitude together.
Position vector
\[\overrightarrow{OA}=\mathbf a\]Vector between points
\[\overrightarrow{AB}=\mathbf b-\mathbf a\]Components
\[\mathbf v=p\mathbf i+q\mathbf j=\begin{pmatrix}p\\q\end{pmatrix}\]Magnitude / unit vector
\[|\mathbf v|=\sqrt{p^2+q^2},\qquad \hat{\mathbf v}=\frac{\mathbf v}{|\mathbf v|}\]Component arithmetic
If \(\mathbf a=4\mathbf i+3\mathbf j\) and \(\mathbf b=\mathbf i+7\mathbf j\), then \[\overrightarrow{AB}=\mathbf b-\mathbf a=-3\mathbf i+4\mathbf j,\qquad |AB|=5.\] A unit vector parallel to \(AB\) is \(-\frac35\mathbf i+\frac45\mathbf j\).2. Geometry by comparing vector paths
Vector geometry often avoids coordinates entirely; the algebra itself proves the geometry.
- Assign simple vectors to convenient sides, often \(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OB}=\mathbf b\).
- Write every target vector by following a path and adding/subtracting vectors.
- Use ratio information to scale a side vector.
- For parallel/collinear results, show one vector is a scalar multiple of another.
- If two non-parallel basis vectors \(\mathbf a,\mathbf b\) are equal in two expressions, equate their coefficients.
Collinearity
If \(\overrightarrow{PQ}=3\mathbf a-6\mathbf b\) and \(\overrightarrow{PR}=-\mathbf a+2\mathbf b\), then \[\overrightarrow{PQ}=-3\overrightarrow{PR},\] so \(P,Q,R\) are collinear.Exam tip. A vector proof needs a conclusion. After obtaining a scalar multiple, explicitly state “therefore the vectors are parallel / the points are collinear”.
3. Internal division and ratios
Ratio points are especially common in triangle and parallelogram proofs.
Section formula in vectors
\[AP:PB=m:n\Rightarrow \overrightarrow{OP}=\frac{n\mathbf a+m\mathbf b}{m+n}\]Midpoint
The midpoint \(M\) of \(AB\) is the special case \(1:1\): \[\overrightarrow{OM}=\frac{\mathbf a+\mathbf b}{2}.\]Common mistake. In the section formula, the coefficient of \(\mathbf a\) is the opposite segment ratio \(n\), and the coefficient of \(\mathbf b\) is \(m\). Derive it from a path if unsure.
4. Area arguments with vectors
Area questions usually combine a vector-derived ratio with elementary triangle-area reasoning.
Shared-altitude ratio
If two triangles have bases on the same straight line and share the same perpendicular height, their areas are in the ratio of their base lengths. Vector ratio questions often use this fact after a line-segment ratio has been established.Exam tip. Do not assume area scales linearly under every length ratio. Similar figures give square scaling; triangles with a common height give base-length scaling.