變換、矩陣與向量：練習題

If matrix \( \mathbf{M} = \begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix} \), which matrix represents \( 3\mathbf{M} \)?

A shape with an area of \( 5 \text{ cm}^2 \) is transformed by an enlargement with a scale factor of \( -3 \). What is the area of the enlarged shape?

In parallelogram \( ABCD \), \( \vec{AB} = \mathbf{p} \) and \( \vec{AD} = \mathbf{q} \). Point \( E \) lies on the side \( BC \) such that \( BE:EC = 2:1 \). Find the vector \( \vec{AE} \) in terms of \( \mathbf{p} \) and \( \mathbf{q} \).

The point \( P(3, 5) \) is reflected in the line \( x = 0 \) (the y-axis). What are the coordinates of the image point \( P' \)?

Calculate the product of the matrices \( \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 4 \\ -1 \end{pmatrix} \).

A point \( A \) with coordinates \( (5, -3) \) is reflected in the \( y \)-axis. State the coordinates of the image point \( A' \).

In triangle \( XYZ \), \( \vec{XY} = \mathbf{u} \) and \( \vec{XZ} = \mathbf{v} \). Point \( M \) lies on the line segment \( YZ \) such that \( YM:MZ = 3:2 \). Express the vector \( \vec{XM} \) in terms of \( \mathbf{u} \) and \( \mathbf{v} \).

The matrix \( \mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \) represents a reflection in the line \( y = x \). The matrix \( \mathbf{N} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \) represents a rotation of \( 90^\circ \) anticlockwise about the origin. Find the matrix \( \mathbf{P} \) that represents the transformation \( \mathbf{M} \) followed by \( \mathbf{N} \), and describe the geometric transformation \( \mathbf{P} \) represents.

The matrix M represents a reflection in the line \( y = x \).
The matrix N represents a rotation of \( 90^\circ \) anticlockwise about the origin \( (0,0) \).

(a) Write down the matrix M.
(b) Find the coordinates of the image of the point \( (3, -2) \) under the transformation represented by matrix M.
(c) Work out the single matrix MN that represents the rotation N followed by the reflection M. Describe fully the single transformation represented by matrix MN.

In triangle \( OAB \), \( \vec{OA} = \mathbf{a} \) and \( \vec{OB} = \mathbf{b} \).
Point \( M \) lies on \( OA \) such that \( OM : MA = 2 : 1 \).
Point \( N \) is the midpoint of \( OB \).
The lines \( AN \) and \( BM \) intersect at the point \( X \).

(a) Express the vectors \( \vec{AN} \) and \( \vec{BM} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \).
(b) By using the fact that \( X \) lies on both \( AN \) and \( BM \), find the values of \( h \) and \( k \) such that \( \vec{OX} = h\mathbf{a} + k\mathbf{b} \).