Scalar and vector products in vectors：練習問題

Two vectors \( \mathbf{a} = \mathbf{i} + k\mathbf{j} + 2\mathbf{k} \) and \( \mathbf{b} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k} \) are known to be perpendicular. Find the value of the scalar constant \( k \).

The vectors \( \mathbf{u} \) and \( \mathbf{v} \) satisfy \( |\mathbf{u}| = 2 \), \( |\mathbf{v}| = 5 \) and the scalar product \( \mathbf{u} \cdot \mathbf{v} = -6 \). Calculate the value of the magnitude \( |\mathbf{u} - 2\mathbf{v}| \).

If \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are unit vectors such that \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0} \), calculate the value of the expression \( \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a} \).

Calculate the scalar product \( \mathbf{a} \cdot \mathbf{b} \) where \( \mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix} \).

Given that \( |\mathbf{a}| = 3 \), \( |\mathbf{b}| = 4 \), and the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \) is \( 60^\circ \), find the magnitude of the vector product \( |\mathbf{a} \times \mathbf{b}| \).

Given a point A with position vector \( \mathbf{a} \) and a line \( L \) passing through the origin in the direction of the unit vector \( \hat{\mathbf{n}} \), state the geometrical significance of the magnitude of the vector product \( |\mathbf{a} \times \hat{\mathbf{n}}| \).