📐 Warp & Weft: Mastering Shear and Stretch!

Transformations don’t stop at rotations and reflections—**shears and stretches** let us reshape figures while preserving parallelism or one set of dimensions. A solid grasp of these skills boosts your marks in coordinate geometry and matrix questions, and equips you for higher-level vector work.

**Shear**: Slides layers parallel to a chosen axis. One axis (the invariant line) stays put, the other shifts by a factor $k$ .
**Stretch**: Enlarges or shrinks only in one direction. A stretch with scale factor $s$ about the origin multiplies all $x$ – or $y$ –coordinates by $s$ .

Worked Example: X-Axis Shear of a Point

Problem: A shear parallel to the

x

-axis with the

y

-axis invariant and shear factor

k=2

is applied to point

P(3,2)

. Find

P'

and state the transformation matrix.

Step 1: The matrix for an $x$ -axis shear of factor $k$ is
$\begin{pmatrix}1 & k\\0 & 1\end{pmatrix}.$
Step 2: Substitute $k=2$ →
$S=\begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}.$
Step 3: Multiply by the coordinate column
$S\begin{pmatrix}3\\2\end{pmatrix}=\begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}\begin{pmatrix}3\\2\end{pmatrix}=\begin{pmatrix}1\cdot3+2\cdot2\\0\cdot3+1\cdot2\end{pmatrix}=\begin{pmatrix}7\\2\end{pmatrix}.$
Step 4: Therefore $P'(7,2)$ . The $y$ -coordinate is unchanged, while the $x$ -coordinate increased by $2 \times y$ —exactly as a shear dictates.