Vectors and transformation geometry：練習問題

Find the magnitude (modulus) of the vector \(\mathbf{v} = \begin{pmatrix} 8 \\ -6 \end{pmatrix}\).

In triangle \(OAB\), \(\vec{OA} = \mathbf{a}\) and \(\vec{OB} = \mathbf{b}\). Point \(M\) lies on the line segment \(AB\) such that the ratio \(AM:MB = 1:2\). Express the position vector \(\vec{OM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

A transformation \(T\) consists of a reflection in the line \(y = x\) followed by a rotation of \(90^{\circ}\) anticlockwise about the origin. Which matrix represents the single transformation \(T\)?

Given the vectors \(\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\), calculate the resultant vector \(2\mathbf{a} + \mathbf{b}\).

Given the vectors \(\mathbf{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -2 \\ 7 \end{pmatrix}\), calculate the magnitude (modulus) of the vector \(3\mathbf{a} - \mathbf{b}\).

In the triangle \(ABC\), \(\vec{AB} = \mathbf{p}\) and \(\vec{BC} = \mathbf{q}\). Express the vector \(\vec{CA}\) in terms of \(\mathbf{p}\) and \(\mathbf{q}\).

The transformation \(\mathbf{T}\) is a rotation of \(90^\circ\) anticlockwise about the origin followed by a translation by the vector \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\). Find the image of the point \((1, 4)\) under \(\mathbf{T}\).