Vectors in two dimensions：練習問題

Given that \( \mathbf{p} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} \) and \( \mathbf{q} = \begin{pmatrix} -1 \\ 4 \end{pmatrix} \), find the column vector \( 2\mathbf{p} - \mathbf{q} \).

Given the vector \(\mathbf{v} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}\), find the magnitude of \(\mathbf{v}\), denoted by \(|\mathbf{v}|\).

In a parallelogram OABC, \( \vec{OA} = \mathbf{a} \) and \( \vec{OC} = \mathbf{c} \). Point P lies on the diagonal AC such that \( AP : PC = 1 : 2 \). Point Q is the midpoint of the side BC. Express the vector \( \vec{PQ} \) in terms of \( \mathbf{a} \) and \( \mathbf{c} \).

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

In triangle \(OAB\), \(\vec{OA} = 6\mathbf{a}\) and \(\vec{OB} = 4\mathbf{b}\). The point \(X\) lies on \(OA\) such that \(OX : XA = 2 : 1\), and the point \(Y\) lies on \(AB\) such that \(\vec{AY} = \frac{1}{3}\vec{AB}\). The lines \(OY\) and \(XB\) intersect at the point \(G\). Given that \(\vec{OG} = h\vec{OY}\) and \(\vec{XG} = k\vec{XB}\), find the value of \(h\).

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

In triangle \( OAB \), \( \vec{OA} = \mathbf{a} \) and \( \vec{OB} = \mathbf{b} \). The point \( X \) lies on the line \( OA \) produced such that \( OA:AX = 2:1 \), and point \( Y \) lies on \( AB \) such that \( AY = 2YB \). Express the vector \( \vec{XY} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \).

In triangle \( OAB \), \( \vec{OA} = \mathbf{a} \) and \( \vec{OB} = \mathbf{b} \). The point \( P \) lies on \( OA \) such that \( OP = \frac{2}{3}OA \). The point \( Q \) is the midpoint of \( AB \). The line \( PQ \) is produced to \( R \) such that \( PQ = QR \).
(a) Express \( \vec{AB} \) and \( \vec{OQ} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \).
(b) Express \( \vec{PQ} \) and \( \vec{OR} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \).
(c) Show that \( O, B, \) and \( R \) are collinear and find the ratio \( OB:BR \).

In triangle \( OAB \), \( \vec{OA} = \mathbf{a} \) and \( \vec{OB} = \mathbf{b} \). Point \( M \) lies on \( OA \) such that \( OA = 3OM \) and \( N \) is the midpoint of \( AB \). The lines \( OB \) and \( MN \) are produced to meet at point \( P \).
(a) Express \( \vec{AB} \) and \( \vec{MN} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \).
(b) Given that \( \vec{MP} = k\vec{MN} \), express \( \vec{OP} \) in terms of \( k, \mathbf{a} \) and \( \mathbf{b} \).
(c) Use the fact that \( P \) lies on the line \( OB \) to find the value of \( k \) and hence express \( \vec{OP} \) in terms of \( \mathbf{b} \) only.