Vectors：練習問題

Given the vectors \(\mathbf{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\), find the vector \(2\mathbf{a} - \mathbf{b}\) as a column vector.

In triangle \(OAB\), \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). The point \(M\) is the midpoint of the line segment \(AB\). Find the vector \(\overrightarrow{OM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

Given the column vector \(\mathbf{v} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}\), calculate the magnitude (modulus) of the vector \(\mathbf{v}\).

In a diagram, \(\vec{OP} = \bf{a}\) and \(\vec{OQ} = \bf{b}\). Point \(R\) lies on the line segment \(PQ\) such that the ratio \(PR:RQ = 2:3\). Express \(\vec{OR}\) in terms of \(\bf{a}\) and \(\bf{b}\).

\(OABC\) is a parallelogram where \(\vec{OA} = \bf{a}\) and \(\vec{OC} = \bf{c}\). Point \(M\) is the midpoint of the side \(AB\). Express the vector \(\vec{OM}\) in terms of \(\bf{a}\) and \(\bf{c}\).

In the diagram, \(OABC\) is a trapezium where \(OA\) is parallel to \(CB\).
\(\vec{OA} = 2\mathbf{a}\) and \(\vec{OC} = \mathbf{c}\).
The length of \(CB\) is three times the length of \(OA\).
\(M\) is the midpoint of \(OC\) and \(N\) is the point on \(AB\) such that \(AN:NB = 1:2\).

(a) Express \(\vec{CB}\) in terms of \(\mathbf{a}\).
(b) Find \(\vec{MN}\) in terms of \(\mathbf{a}\) and \(\mathbf{c}\). Simplify your answer.

The vector \(\mathbf{a}\) is given by \(\binom{3}{-2}\) and the vector \(\mathbf{b}\) is given by \(\binom{-1}{4}\).

(a) Calculate the vector \(2\mathbf{a} - 3\mathbf{b}\) as a column vector.
(b) Calculate the magnitude of the vector \(2\mathbf{a} - 3\mathbf{b}\). Give your answer as a surd in its simplest form.