Further Vectors: แบบฝึกหัด

Determine the value of the constant \( k \) for which the vectors \( \mathbf{u} = \begin{pmatrix} 2 \\ k \\ -3 \end{pmatrix} \) and \( \mathbf{v} = \begin{pmatrix} k+1 \\ 4 \\ 2 \end{pmatrix} \) are perpendicular.

Find the coordinates of the point where the line with equation \( \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} \) intersects the plane with Cartesian equation \( x + 2y - z = 5 \).

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

A plane has the vector equation \( \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix} \).
Find the equivalent Cartesian equation of this plane.

The vertices of a tetrahedron are defined by the points \( O(0,0,0) \), \( A(2,1,0) \), \( B(1,3,0) \), and \( C(1,1,4) \).
Using the scalar triple product, calculate the volume of the tetrahedron.

Find the value of the constant \(k\) for which the vectors \(\mathbf{u} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} k \\ 4 \\ 5 \end{pmatrix}\) are perpendicular.

A plane has the vector equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}\). Find the Cartesian equation of this plane in the form \(ax + by + cz = d\).

Two skew lines have equations \(L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\) and \(L_2: \mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}\). Given that a mutual perpendicular to both lines is \(\mathbf{n} = \mathbf{i} + 2\mathbf{j} - 5\mathbf{k}\), find the shortest distance between these two lines.