Scalar product and vector product：練習問題

Let \(\mathbf{u}\) and \(\mathbf{v}\) be two vectors such that \(|\mathbf{u}| = 2\), \(|\mathbf{v}| = 5\) and \(|\mathbf{u} \times \mathbf{v}| = 8\). Find the value of \(|\mathbf{u} \cdot \mathbf{v}|\).

Given that \(\mathbf{u}\) is a unit vector, find the value of the scalar product \((3\mathbf{u}) \cdot (4\mathbf{u})\).

If \(\mathbf{a}\) and \(\mathbf{b}\) are vectors such that \(\mathbf{a} \times \mathbf{b} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}\), find the vector expression for \((2\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b})\).

Given that \(\mathbf{u} = 2\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}\) and \(\mathbf{v} = 4\mathbf{i} - \mathbf{j} + k\mathbf{k}\), find the value of the constant \(k\) such that \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular.

Consider three points \(A(1, 0, 2)\), \(B(3, -1, 0)\), and \(C(k, 2, 4)\) in a three-dimensional rectangular coordinate system, where \(k\) is a constant.

(a) Find the vectors \(\vec{AB}\) and \(\vec{AC}\) in terms of \(k\).
(b) If \(\vec{AB}\) is perpendicular to \(\vec{AC}\), find the value of \(k\).
(c) Using the value of \(k\) found in (b),
    (i) find the vector product \(\vec{AB} \times \vec{AC}\);
    (ii) find the area of triangle \(ABC\).

Let \(\mathbf{a}\) and \(\mathbf{b}\) be two unit vectors in three-dimensional space such that the angle between them is \(\theta\), where \(0 < \theta < \pi\).

(a) Show that \(|\mathbf{a} - \mathbf{b}| = 2\sin\frac{\theta}{2}\).

(b) Let \(\mathbf{c} = (\mathbf{a} \cdot \mathbf{b})(\mathbf{a} \times \mathbf{b})\).
(i) Explain why \(\mathbf{c}\) is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).
(ii) Show that \(|\mathbf{c}| = \frac{1}{2}|\sin 2\theta|\).

(c) Suppose \(|\mathbf{c}| = \frac{1}{4}\).
(i) Find all possible values of \(\theta\).
(ii) For the smallest value of \(\theta\) obtained in (c)(i), find the volume of the parallelepiped formed by the vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{a} \times \mathbf{b}\).