Line intersection: show lines are skew

7. Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } - 4 \\ 1 \\ 3 \end{array} \right)\) and \(\left( \begin{array} { c } - 3 \\ 6 \\ 1 \end{array} \right)\) respectively.

1. Find a vector equation for the line \(l _ { 1 }\) which passes through \(A\) and \(B\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } 3 \\ - 7 \\ 9 \end{array} \right) + t \left( \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right)$$
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
3. Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).

+ s \left( \begin{array} { l } 3
2
1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 1
0
0 \end{array} \right) + t \left( \begin{array} { r } 0
1
- 1 \end{array} \right)$$ respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
3. The point \(A\) lies on \(l _ { 1 }\) and \(O A\) is perpendicular to \(l _ { 1 }\), where \(O\) is the origin. Find the position vector of \(A\). 6 Find the coefficient of \(x ^ { 2 }\) in the expansion in ascending powers of \(x\) of $$\sqrt { \frac { 1 + a x } { 4 - x } } ,$$ giving your answer in terms of \(a\).

Two lines have equations \(\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}\).

1. Show that the lines are skew. [5]
2. Find the acute angle between the directions of the two lines. [3]

The line \(L_1\) passes through the points \((2, -3, 1)\) and \((-1, -2, -4)\). The line \(L_2\) passes through the point \((3, 2, -9)\) and is parallel to the vector \(\mathbf{4i} - \mathbf{4j} + \mathbf{5k}\).

1. Find an equation for \(L_1\) in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [2]
2. Prove that \(L_1\) and \(L_2\) are skew. [5]

Two lines have vector equations $$\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 4\mathbf{k} + \lambda(3\mathbf{i} + \mathbf{j} + a\mathbf{k})$$ and $$\mathbf{r} = -8\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \mu(\mathbf{i} - 2\mathbf{j} - \mathbf{k}),$$ where \(a\) is a constant.

1. Given that the lines are skew, find the value that \(a\) cannot take. [6]
2. Given instead that the lines intersect, find the point of intersection. [2]

Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ 6 \\ 1 \end{pmatrix}\) respectively.

1. Find a vector equation for the line \(l_1\) which passes through \(A\) and \(B\). [2]

The line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ -7 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}.$$

1. Show that lines \(l_1\) and \(l_2\) do not intersect. [5]
2. Find the position vector of the point \(C\) on \(l_2\) such that \(\angle ABC = 90°\). [6]

The lines \(L_1\) and \(L_2\) have vector equations $$L_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 10 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -5 \\ 4 \end{pmatrix}$$ $$L_2 : \mathbf{r} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$$

1. Show that \(L_1\) and \(L_2\) are perpendicular. [2]
2. Show that \(L_1\) and \(L_2\) are skew lines. [3] The point \(A\) with position vector \(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) lies on \(L_2\) and the point \(X\) lies on \(L_1\) such that \(\overrightarrow{AX}\) is perpendicular to \(L_1\)
3. Find the position vector of \(X\). [5]
4. Find \(|\overrightarrow{AX}|\) [2] The point \(B\) (distinct from \(A\)) also lies on \(L_2\) and \(|\overrightarrow{BX}| = |\overrightarrow{AX}|\)
5. Find the position vector of \(B\). [5]
6. Find the cosine of angle \(AXB\). [2]