Cambridge International A Level · Mathematics - Further (9231)
3 multiple-choice questions marked as you go, and 3 written questions with worked solutions. All on Vectors.
Find the vector equation of the common perpendicular to the skew lines \(L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + t \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}\) and \(L_2: \mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\).
Find the acute angle, correct to one decimal place, between the plane \(\Pi: x - 2y + 2z = 5\) and the line \(L: \mathbf{r} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}\).
Find the Cartesian equation of the plane that passes through the point \(A(1, 2, 3))\) and contains the line \(\mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} + t \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\).
The line \(L\) has equation \(\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\). The plane \(\Pi\) has equation \(x - y + 2z = 5\). Find the acute angle between the line \(L\) and the plane \(\Pi\).
Write your answer out first, then check it against the worked solution.
A plane has the Cartesian equation \(2x - y + 3z = 7\).
(a) Write down a normal vector to the plane.
(b) Determine whether the point \(A(1, 2, 3)\) lies on the plane.
(c) Find the perpendicular distance from the origin to the plane.
Write your answer out first, then check it against the worked solution.
A plane \(\Pi\) has the vector equation
\(\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}\).
(a) Find the Cartesian equation of the plane \(\Pi\).
(b) Calculate the perpendicular distance from the point \(P(4, 5, 1)\) to the plane \(\Pi\).
Write your answer out first, then check it against the worked solution.
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