Further Vectors: แบบฝึกหัด

Determine the value of the constant \(k\) such that the line with direction vector \(\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) is perpendicular to the line with direction vector \(\begin{pmatrix} k \\ 4 \\ 2 \end{pmatrix}\).

Find the shortest distance from the point \((1, -2, 3)\) to the plane with equation \(2x - 3y + 6z = 10\).

A tetrahedron has vertices at the points \(O(0, 0, 0)\), \(A(2, 0, 0)\), \(B(1, 3, 0)\), and \(C(1, 1, 4)\).
Calculate the volume of the tetrahedron.

Find the acute angle between the line \(\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\) and the plane \(x + z = 5\).

Find the shortest distance between the skew lines \(L_1\) and \(L_2\) with equations:
\(L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\)
\(L_2: \mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\)

Find the value of the constant \(k\) such that the vectors \(\mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}\) and \(\mathbf{b} = k\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\) are perpendicular.

Find the coordinates of the point of intersection between the line with vector equation \(\mathbf{r} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}\) and the plane with cartesian equation \(2x - y + z = 12\).

A point \(P\) has position vector \(\mathbf{b} = \begin{pmatrix} 1 \\ 5 \\ -2 \end{pmatrix}\) and a plane is defined by the equation \(\mathbf{r} \cdot \begin{pmatrix} 3 \\ -4 \\ 12 \end{pmatrix} = 10\). Use the formula \(D = \frac{|\mathbf{b} \cdot \mathbf{n} - p|}{|\mathbf{n}|}\) to calculate the shortest distance from point \(P\) to the plane.

The line \(L\) has vector equation \(\mathbf{r} = \begin{pmatrix} 1 \\\\ -2 \\\\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\\\ 1 \\\\ -1 \end{pmatrix}\) and the plane \(\Pi\) has cartesian equation \(x - 2y + 2z = 7\).
(a) Find the coordinates of the point \(P\) where the line \(L\) intersects the plane \(\Pi\).
(b) Calculate the acute angle between the line \(L\) and the plane \(\Pi\), giving your answer in degrees to one decimal place.

The lines \(L_1\) and \(L_2\) are defined by the following vector equations:
\(L_1: \mathbf{r} = \begin{pmatrix} 1 \\\\ 0 \\\\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\\\ 2 \\\\ -1 \end{pmatrix}\)
\(L_2: \mathbf{r} = \begin{pmatrix} 2 \\\\ 1 \\\\ 0 \end{pmatrix} + \mu \begin{pmatrix} 0 \\\\ 1 \\\\ 1 \end{pmatrix}\)
(a) Show that the lines \(L_1\) and \(L_2\) are skew.
(b) Use the vector product to find a vector \(\mathbf{n}\) that is perpendicular to both \(L_1\) and \(L_2\).
(c) Hence, determine the shortest distance between the lines \(L_1\) and \(L_2\), giving your answer in exact form.