Vectors：練習問題

Given the vector \(\mathbf{a} = 3\mathbf{i} - 2\mathbf{j} + 6\mathbf{k}\), find its magnitude, \(|\mathbf{a}|\).

Two vectors are given by \(\mathbf{a} = 2\mathbf{i} + \mathbf{j} - 3\mathbf{k}\) and \(\mathbf{b} = \mathbf{i} - 4\mathbf{j} + x\mathbf{k}\). If \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular, find the value of \(x\).

A line \(L\) passes through points \(A(1, 2, 3)\) and \(B(4, -1, 0)\). Which of the following is a vector equation of line \(L\)?

The position vectors of points \(A\) and \(B\) are \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}\) and \(\mathbf{b} = 6\mathbf{i} - \mathbf{j} + 7\mathbf{k}\) respectively. Point \(P\) lies on the line segment \(AB\) such that \(AP:PB = 1:3\). Find the position vector of \(P\).

The vectors \(\mathbf{a}\) and \(\mathbf{b}\) are defined as \(\mathbf{a} = 5\mathbf{i} - 2\mathbf{j} + \mathbf{k}\) and \(\mathbf{b} = 2\mathbf{i} + k\mathbf{j} - 4\mathbf{k}\). Given that \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular, find the value of the constant \(k\).

Given vectors \(\mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}\) and \(\mathbf{b} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\), calculate their scalar product \(\mathbf{a} \cdot \mathbf{b}\). Are the vectors perpendicular?

Points A and B have position vectors \(\mathbf{a} = 2\mathbf{i} + \mathbf{j}\) and \(\mathbf{b} = 5\mathbf{i} - 2\mathbf{j}\) respectively. Find the position vector of the midpoint of AB.

Relative to a fixed origin \(O\), the position vectors of points \(A\) and \(B\) are given by:
\(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}\)
\(\mathbf{b} = \mathbf{i} + \mathbf{j} + \lambda\mathbf{k}\)
where \(\lambda\) is a constant.
(a) Find the vector \(\vec{AB}\) in terms of \(\lambda\).
(b) Given that the magnitude of \(\vec{AB}\) is \(3\) units, find the two possible values of \(\lambda\).
(c) Using the smaller value of \(\lambda\) found in part (b), calculate the scalar product \(\mathbf{a} \cdot \mathbf{b}\).

Relative to a fixed origin \(O\), the line \(L\) passes through the points \(A(1, 0, 3)\) and \(B(2, 4, -1)\).
(a) Find a vector equation for line \(L\).
(b) The point \(P\) has position vector \(5\mathbf{i} + 2\mathbf{j} + \mathbf{k}\). Find the coordinates of the point \(D\) on \(L\) such that \(PD\) is perpendicular to \(L\).