Show lines intersect and find intersection point

The line \(l_1\) passes through the points \(A\) and \(B\) with position vectors \((\mathbf{3i} + \mathbf{6j} - \mathbf{8k})\) and \((\mathbf{8j} - \mathbf{6k})\) respectively, relative to a fixed origin.

1. Find a vector equation for \(l_1\). [2]

The line \(l_2\) has vector equation $$\mathbf{r} = (-\mathbf{2i} + \mathbf{10j} + \mathbf{6k}) + \mu(\mathbf{7i} - \mathbf{4j} + \mathbf{6k}),$$ where \(\mu\) is a scalar parameter.

1. Show that lines \(l_1\) and \(l_2\) intersect. [4]
2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [2]

The point \(C\) lies on \(l_2\) and is such that \(AC\) is perpendicular to \(AB\).

1. Find the position vector of \(C\). [6]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.

1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]

A straight road passes through villages at the points \(A\) and \(B\) with position vectors \((9\mathbf{i} - 8\mathbf{j} + 2\mathbf{k})\) and \((4\mathbf{j} + \mathbf{k})\) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf{r} = (2\mathbf{i} + 16\mathbf{j} - \mathbf{k}) + t(-5\mathbf{i} + 3\mathbf{j}),$$ where \(t\) is a scalar parameter.

1. Find the position vector of \(C\). [5]

Given that 1 unit on each coordinate axis represents 200 metres,

1. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\). [4]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of \(P\). [2]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [2]

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations \begin{align} l_1: \quad \mathbf{r} &= (-9\mathbf{i} + 10\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})
l_2: \quad \mathbf{r} &= (3\mathbf{i} + \mathbf{j} + 17\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k}) \end{align} where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l_1\) and \(l_2\) meet and find the position vector of their point of intersection. [6]
2. Show that \(l_1\) and \(l_2\) are perpendicular to each other. [2]

The point \(A\) has position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\).

1. Show that \(A\) lies on \(l_1\). [1]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -1 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ P is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of P. [3]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]

Q is a point on \(l_1\) which is 12 metres away from P. R is the point on \(l_2\) such that QR is perpendicular to \(l_1\).

1. Determine the length QR. [2]

The equations of two lines are \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \lambda(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) and \(\mathbf{r} = 6\mathbf{i} + 8\mathbf{j} + \mathbf{k} + \mu(\mathbf{i} + 4\mathbf{j} - 5\mathbf{k})\).

1. Show that these lines meet, and find the coordinates of the point of intersection. [5]
2. Find the acute angle between these lines. [3]

The points \(A\) and \(B\) have position vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\) respectively, relative to the origin \(O\). The point \(C\) is on the line \(OA\) extended so that \(\overrightarrow{AC} = 2\overrightarrow{OA}\) and the point \(D\) is on the line \(OB\) extended so that \(\overrightarrow{BD} = 3\overrightarrow{OB}\). The point \(X\) is such that \(OCXD\) is a parallelogram.

1. Show that a vector equation of the line \(AX\) is \(\mathbf{r} = \mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(5\mathbf{i} + 7\mathbf{k})\) and find an equation of the line \(CD\) in a similar form. [5]
2. Prove that the lines \(AX\) and \(CD\) intersect and find the position vector of their point of intersection. [4]