Vectors in two dimensions：練習問題

If \(\mathbf{p} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}\), find the vector result of \(\mathbf{p} + 2\mathbf{q}\).

A boat can travel at 10 m s\(^{-1}\) in still water. The boat aims to cross a river of width 120 m where the current flows at 6 m s\(^{-1}\). The boat is steered at an angle upstream so that it travels directly across the river to a point on the opposite bank. Find the time taken to cross the river.

The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are \(\vec{OA} = \mathbf{a}\) and \(\vec{OB} = \mathbf{b}\). Express the vector \(\vec{AB}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

The position vectors of points \( P \) and \( Q \) relative to an origin \( O \) are \( \vec{OP} = \binom{2}{k} \) and \( \vec{OQ} = \binom{5}{2} \). Given that the magnitude of the vector \( \vec{PQ} \) is \( \sqrt{13} \), find the possible values of \( k \).

Given the vector a = \(3\mathbf{i} - 4\mathbf{j}\), calculate the magnitude \(|\mathbf{a}|\).

Relative to an origin \(O\), the position vectors of points \(P\) and \(Q\) are \(\vec{OP} = 3\mathbf{i} + 4\mathbf{j}\) and \(\vec{OQ} = 11\mathbf{i} - 2\mathbf{j}\). Point \(R\) lies on \(PQ\) such that \(PR:RQ = 3:1\). Find the unit vector in the direction of \(\vec{OR}\).

In triangle \(ABC\), the position vectors of points \(A\) and \(B\) relative to an origin \(O\) are given by \(\vec{OA} = 5\mathbf{i} + 2\mathbf{j}\) and \(\vec{OB} = 11\mathbf{i} + 10\mathbf{j}\). Point \(P\) lies on the line segment \(AB\) such that \(AP:PB = 1:2\). Point \(Q\) is such that \(\vec{OQ} = k\mathbf{i} + (k + 5)\mathbf{j}\), where \(k\) is a constant.

(a) Find the position vector of \(P\).
(b) Given that the magnitude of vector \(\vec{PQ}\) is \(\sqrt{13}\), find the two possible values of \(k\).
(c) For the larger value of \(k\) found in part (b), find the unit vector in the direction of \(\vec{PQ}\).

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(\vec{OA} = 6〈〉\mathbf{i} - 2〈〉\mathbf{j}\) and \(\vec{OB} = k〈〉\mathbf{i} + 13〈〉\mathbf{j}\). Point \(C\) lies on \(AB\) such that \(\vec{AC} = \frac{2}{5} 〈〉\vec{AB}\).
(a) Express \(\vec{OC}\) in terms of \(k\), \(〈〉\mathbf{i}\) and \(〈〉\mathbf{j}\).
(b) Given that \(O\), \(C\) and a point \(D\) with position vector \(14〈〉\mathbf{i} + 14〈〉\mathbf{j}\) are collinear, find the value of the constant \(k\).
(c) For this value of \(k\), find the magnitude of \(〈〉\vec{AB}\) and the unit vector in the direction of \(〈〉\vec{AB}\).