向量加減與純量乘法：練習題

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

The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are \(\mathbf{a} = 2\mathbf{i} + 5\mathbf{j}\) and \(\mathbf{b} = 7\mathbf{i} - 5\mathbf{j}\) respectively. The point \(P\) lies on the line segment \(AB\) such that \(AP:PB = 2:3\). Find the position vector of \(P\).

In triangle \(OAB\), \(\vec{OA} = \mathbf{a}\) and \(\vec{OB} = \mathbf{b}\). The point \(P\) lies on \(AB\) such that \(AP:PB = 3:1\). The point \(M\) is the midpoint of \(OA\). The lines \(OP\) and \(BM\) intersect at the point \(X\). Given that \(\vec{OX} = k\vec{OP}\), find the value of \(k\).

Points \(A\) and \(B\) have position vectors \(\vec{OA} = 4\mathbf{i} + 3\mathbf{j}\) and \(\vec{OB} = 10\mathbf{i} - 5\mathbf{j}\) respectively. Find the unit vector in the direction of \(\vec{AB}\).

Given that \(\mathbf{a}\) and \(\mathbf{b}\) are non-parallel vectors and that \((2x + y - 5)\mathbf{a} + (x - 3y + 8)\mathbf{b} = \mathbf{0}\), find the values of \(x\) and \(y\).

Given that vectors \(\mathbf{a}\) and \(\mathbf{b}\) are non-parallel and that \((2k - 1)\mathbf{a} + 3\mathbf{b} = 5\mathbf{a} + (m + 2)\mathbf{b}\), find the values of the scalars \(k\) and \(m\).

The points \(A\) and \(B\) have position vectors \(4\mathbf{i} - 3\mathbf{j}\) and \(k\mathbf{i} + 5\mathbf{j}\) respectively. Given that the magnitude of the vector \(\vec{AB}\) is 10 units, find the two possible values of the constant \(k\).

Point \(C\) lies on the line segment \(AB\) such that \(\vec{OC} = \frac{3}{7}\vec{OA} + \frac{4}{7}\vec{OB}\), where \(O\) is the origin. Find the ratio \(AC:CB\).

Relative to a fixed origin \(O\), the position vectors of the points \(A\) and \(B\) are given by \(\mathbf{a} = 4\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{b} = -2\mathbf{i} + 9\mathbf{j}\) respectively.

(a) Find the vector \(\vec{AB}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\).
(b) Calculate the magnitude of \(\vec{AB}\).

The position vectors of points \(P\) and \(Q\) relative to an origin \(O\) are \(\mathbf{p} = 2\mathbf{a} + 5\mathbf{b}\) and \(\mathbf{q} = 7\mathbf{a} - \mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are non-parallel vectors. The point \(R\) lies on the line \(PQ\) such that \(PR:RQ = 2:3\).

(a) Express the position vector of \(R\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
(b) Given that the position vector of point \(S\) is \(k\mathbf{a} + 1.4\mathbf{b}\) and that \(O\), \(R\), and \(S\) are collinear, find the value of the constant \(k\).