Vector geometry：練習問題

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

Find the magnitude of the vector \(\vec{PQ}\) where the coordinates of the points are \(P(1, 2)\) and \(Q(4, 6)\).

Points \(X, Y,\) and \(Z\) are located such that \(\vec{XY} = \begin{pmatrix} 2 \\ k \end{pmatrix}\) and \(\vec{YZ} = \begin{pmatrix} 3 \\ 6 \end{pmatrix}\). Given that points \(X, Y,\) and \(Z\) lie on a single straight line, determine the value of \(k\).

If the vector \(\mathbf{v} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}\), calculate the components of the vector \(3\mathbf{v}\).

The position vector of point \(A\) is \(\mathbf{a}\) and the position vector of point \(B\) is \(\mathbf{b}\). Point \(M\) is the midpoint of the line segment \(AB\). Express the position vector of \(M\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

In triangle \(OAB\), \(\vec{OA} = \mathbf{a}\) and \(\vec{OB} = \mathbf{b}\). Point \(M\) is the midpoint of \(OA\) and point \(N\) lies on the line segment \(AB\) such that \(AN = \frac{3}{4} AB\). Express the vector \(\vec{MN}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) in its simplest form.

In the diagram, \(\vec{OP} = \mathbf{p}\) and \(\vec{OQ} = \mathbf{q}\). Point \(R\) is defined such that \(\vec{OR} = 3\mathbf{p} + 2\mathbf{q}\). If \(S\) is the midpoint of the line \(PR\), express the position vector \(\vec{OS}\) in terms of \(\mathbf{p}\) and \(\mathbf{q}\).

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

In the parallelogram \(OABC\), \(\vec{OA} = \mathbf{a}\) and \(\vec{OC} = \mathbf{c}\). The point \(M\) is the midpoint of the side \(BC\) and the point \(N\) lies on the diagonal \(AC\) such that \(AN:NC = 2:1\).

(a) Express \(\vec{AC}\) and \(\vec{OM}\) in terms of \(\mathbf{a}\) and \(\mathbf{c}\).

(b) Show that \(\vec{ON} = \frac{1}{3}\mathbf{a} + \frac{2}{3}\mathbf{c}\).

(c) Use your answers to parts (a) and (b) to show that the points \(O, N,\) and \(M\) are collinear and find the ratio \(ON:OM\).