Point on line with condition

9 \includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-4_724_1488_257_330} The diagram shows a cuboid \(O A B C D E F G\) with a horizontal base \(O A B C\) in which \(O A = 4 \mathrm {~cm}\) and \(A B = 15 \mathrm {~cm}\). The height \(O D\) of the cuboid is 2 cm . The point \(X\) on \(A B\) is such that \(A X = 5 \mathrm {~cm}\) and the point \(P\) on \(D G\) is such that \(D P = p \mathrm {~cm}\), where \(p\) is a constant. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.

1. Find the possible values of \(p\) such that angle \(O P X = 90 ^ { \circ }\).
2. For the case where \(p = 9\), find the unit vector in the direction of \(\overrightarrow { X P }\).
3. A point \(Q\) lies on the face \(C B F G\) and is such that \(X Q\) is parallel to \(A G\). Find \(\overrightarrow { X Q }\).

6. Relative to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates ( \(2,1,9 ) , ( 5,2,7 )\) and \(( 4 , - 3,3 )\) respectively. The line \(l\) passes through the points \(A\) and \(B\).

1. Find a vector equation for the line \(l\).
2. Find, in degrees, the acute angle between the line \(I\) and the line \(A C\). The point \(D\) lies on the line \(l\) such that angle \(A C D\) is \(90 ^ { \circ }\)
3. Find the coordinates of \(D\).
4. Find the exact area of triangle \(A D C\), giving your answer as a fully simplified surd.

4. The line \(l _ { 1 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } - 6 \\ 4 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ - 1 \\ 3 \end{array} \right)$$ and the line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } - 6 \\ 4 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { c } 3 \\ - 4 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\) and the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\).

1. Write down the coordinates of \(A\).
2. Find the value of \(\cos \theta\). The point \(X\) lies on \(l _ { 1 }\) where \(\lambda = 4\).
3. Find the coordinates of \(X\).
4. Find the vector \(\overrightarrow { A X }\).
5. Hence, or otherwise, show that \(| \overrightarrow { A X } | = 4 \sqrt { } 26\). The point \(Y\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { Y X }\) is perpendicular to \(l _ { 1 }\),
6. find the length of \(A Y\), giving your answer to 3 significant figures. \section*{LU}

8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 4 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 2 \\ 5 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) The point \(A\) has position vector \(\left( \begin{array} { l } 0 \\ 1 \\ 6 \end{array} \right)\).
2. Show that \(A\) lies on \(l _ { 1 }\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(X\).
3. Write down the coordinates of \(X\).
4. Find the exact value of the distance \(A X\). The distinct points \(B _ { 1 }\) and \(B _ { 2 }\) both lie on the line \(l _ { 2 }\) Given that \(A X = X B _ { 1 } = X B _ { 2 }\)
5. find the area of the triangle \(A B _ { 1 } B _ { 2 }\) giving your answer to 3 significant figures. Given that the \(x\) coordinate of \(B _ { 1 }\) is positive,
6. find the exact coordinates of \(B _ { 1 }\) and the exact coordinates of \(B _ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-28_96_59_2478_1834}

1. With respect to a fixed origin \(O\), the line \(l\) has equation

$$\mathbf { r } = \left( \begin{array} { c }

13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
2
- 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point \(A\) lies on \(l\) and has coordinates ( \(3 , - 2,6\) ).
The point \(P\) has position vector ( \(- p \mathbf { i } + 2 p \mathbf { k }\) ) relative to \(O\), where \(p\) is a constant.
Given that vector \(\overrightarrow { P A }\) is perpendicular to \(l\),

1. find the value of \(p\). Given also that \(B\) is a point on \(l\) such that \(\angle B P A = 45 ^ { \circ }\),
2. find the coordinates of the two possible positions of \(B\).

8. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$\mathbf { r } = \left( \begin{array} { r } 8 \\ 1 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5 \\ 4 \\ 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)

1. Find the coordinates of \(A\). The point \(P\) has position vector \(\left( \begin{array} { l } 1 \\ 5 \\ 2 \end{array} \right)\).
   The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
2. Write down a vector equation for the line \(l _ { 2 }\)
3. Find the exact value of the distance \(A P\). Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be determined. The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\).
4. Find the value of \(\cos \theta\) A point \(E\) lies on the line \(l _ { 2 }\) Given that \(A P = P E\),
5. find the area of triangle \(A P E\),
6. find the coordinates of the two possible positions of \(E\).

6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ 28 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ - 5 \\ 1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ 0 \\ - 4 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(X\).

1. Find the coordinates of the point \(X\).
2. Find the size of the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to 2 decimal places. The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 2 \\ 18 \\ 6 \end{array} \right)\)
3. Find the distance \(A X\), giving your answer as a surd in its simplest form. The point \(Y\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { Y A }\) is perpendicular to the line \(l _ { 1 }\)
4. find the distance \(Y A\), giving your answer to one decimal place. The point \(B\) lies on \(l _ { 1 }\) where \(| \overrightarrow { A X } | = 2 | \overrightarrow { A B } |\).
5. Find the two possible position vectors of \(B\).

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 \(\angle A B C = 90 ^ { \circ }\).

8. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors \(( - 3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } )\) and ( \(7 \mathbf { i } - \mathbf { j } + 12 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 5 \mathbf { j } - 7 \mathbf { k } ) + \mu ( \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } )$$ The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(B C\).
2. Show that one possible position vector for \(C\) is \(( \mathbf { i } + 3 \mathbf { j } )\) and find the other. Assuming that \(C\) has position vector \(( \mathbf { i } + 3 \mathbf { j } )\),
3. find the area of triangle \(A B C\), giving your answer in the form \(k \sqrt { 5 }\).

7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
4. Find the position vector of \(C\).

7 The equation of a straight line \(l\) is $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) .$$ \(O\) is the origin.

1. Find the position vector of the point \(P\) on \(l\) such that \(O P\) is perpendicular to \(l\).
2. A point \(Q\) on \(l\) is such that the length of \(O Q\) is 3 units. Find the two possible position vectors of \(Q\). [3]

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the straight line \(l\).
Line \(l\) passes through the points \(A\) and \(B\).
Relative to a fixed origin \(O\)

• the point \(A\) has position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 8 \mathbf { k }\)
  1. Find \(\overrightarrow { A B }\)

Given that a point \(P\) lies on \(l\) such that $$| \overrightarrow { A P } | = 2 | \overrightarrow { B P } |$$

find the possible position vectors of \(P\).

\(\mathbf { 7 } \quad\) The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 0 \\ - 2 \\ q \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right]\), where \(q\) is an integer. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 8 \\ 3 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { l } 2 \\ 5 \\ 4 \end{array} \right]\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).

1. Show that \(q = 4\) and find the coordinates of \(P\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
3. The point \(A\) lies on the line \(l _ { 1 }\) where \(\lambda = 1\).
   1. Find \(A P ^ { 2 }\).
   2. The point \(B\) lies on the line \(l _ { 2 }\) so that the right-angled triangle \(A P B\) is isosceles. Find the coordinates of the two possible positions of \(B\).

6 The points \(A , B\) and \(C\) have coordinates \(( 3 , - 2,4 ) , ( 1 , - 5,6 )\) and \(( - 4,5 , - 1 )\) respectively. The line \(l\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 7 \\ - 7 \\ 5 \end{array} \right]\).

1. Show that the point \(C\) lies on the line \(l\).
2. Find a vector equation of the line that passes through points \(A\) and \(B\).
3. The point \(D\) lies on the line through \(A\) and \(B\) such that the angle \(C D A\) is a right angle. Find the coordinates of \(D\).
4. The point \(E\) lies on the line through \(A\) and \(B\) such that the area of triangle \(A C E\) is three times the area of triangle \(A C D\). Find the coordinates of the two possible positions of \(E\).

6 The points \(A\) and \(B\) have coordinates \(( 3,2,10 )\) and \(( 5 , - 2,4 )\) respectively.
The line \(l\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ 2 \\ 10 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right]\).

1. Find the acute angle between \(l\) and the line \(A B\).
2. The point \(C\) lies on \(l\) such that angle \(A B C\) is \(90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{fdd3905e-11f7-4b20-adfe-4c686018a221-12_360_339_762_852} Find the coordinates of \(C\).
3. The point \(D\) is such that \(B D\) is parallel to \(A C\) and angle \(B C D\) is \(90 ^ { \circ }\). The point \(E\) lies on the line through \(B\) and \(D\) and is such that the length of \(D E\) is half that of \(A C\). Find the coordinates of the two possible positions of \(E\).
   [0pt] [4 marks]

3. Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf { r } = ( \mathbf { i } + p \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 3 \mathbf { i } - \mathbf { j } + q \mathbf { k } ) ,$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter.
Given that the point \(A\) with coordinates \(( - 5,9 , - 9 )\) lies on \(l\),

1. find the values of \(p\) and \(q\),
2. show that the point \(B\) with coordinates \(( 25 , - 1,11 )\) also lies on \(l\). The point \(C\) lies on \(l\) and is such that \(O C\) is perpendicular to \(l\).
3. Find the coordinates of \(C\).
4. Find the ratio \(A C : C B\) 3. continued

8. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors \(( - 3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } )\) and ( \(7 \mathbf { i } - \mathbf { j } + 12 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 5 \mathbf { j } - 7 \mathbf { k } ) + \mu ( \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } )$$ The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(B C\).
2. Show that one possible position vector for \(C\) is \(( \mathbf { i } + 3 \mathbf { j } )\) and find the other. Assuming that \(C\) has position vector \(( \mathbf { i } + 3 \mathbf { j } )\),
3. find the area of triangle \(A B C\), giving your answer in the form \(k \sqrt { 5 }\).
   8. continued
   8. continued

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 $$\mathbf { r } = \left( \begin{array} { c } 3 \\ - 7 \\ 9 \end{array} \right) + \mu \left( \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right)$$
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 \(\angle A B C = 90 ^ { \circ }\).
   7. continued

7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
4. Find the position vector of \(C\).
   7. continued
   7. continued

10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the position vector of \(P\).
2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). \(Q\) is a point on \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
3. Determine the length \(Q R\). 4 In this question you must show detailed reasoning.
   The distinct numbers \(\omega _ { 1 }\) and \(\omega _ { 2 }\) both satisfy the quadratic equation \(4 x ^ { 2 } + 4 x + 17 = 0\).
4. Write down the value of \(\omega _ { 1 } \omega _ { 2 }\).
5. \(A , B\) and \(C\) are the points on an Argand diagram which represent \(\omega _ { 1 } , \omega _ { 2 }\) and \(\omega _ { 1 } \omega _ { 2 }\). Find the area of triangle \(A B C\). 5 In this question you must show detailed reasoning.
   The equation \(x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
6. Using the identity \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma\) find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
7. Given that \(\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112\) find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 }\) and \(\gamma ^ { 3 }\). 6 Prove by induction that \(n ! \geqslant 6 n\) for \(n \geqslant 4\). 7 A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s \\ t & 0 \end{array} \right)\). Find in terms of \(s\) the matrices which represent each of the shears. 8
8. (a) Find, in terms of \(x\), a vector which is perpendicular to the vectors \(\left( \begin{array} { c } x - 2 \\ 5 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { c } x \\ 6 \\ 2 \end{array} \right)\).
   (b) Find the shortest possible vector of the form \(\left( \begin{array} { l } 1 \\ a \\ b \end{array} \right)\) which is perpendicular to the vectors \(\left( \begin{array} { c } x - 2 \\ 5 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { c } x \\ 6 \\ 2 \end{array} \right)\).
9. Vector \(\mathbf { v }\) is perpendicular to both \(\left( \begin{array} { c } - 1 \\ 1 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { c } 1 \\ p \\ p ^ { 2 } \end{array} \right)\) where \(p\) is a real number. Show that it is impossible for \(\mathbf { v }\) to be perpendicular to the vector \(\left( \begin{array} { c } 1 \\ 1 \\ p - 1 \end{array} \right)\). \section*{OCR} Oxford Cambridge and RSA

9 The points \(A\) and \(B\) lie on the line \(l _ { 1 }\) and have coordinates \(( 2,5,1 )\) and \(( 4,1 , - 2 )\) respectively.

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Find a vector equation of the line \(l _ { 1 }\), with parameter \(\lambda\).
2. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 1 \\ - 3 \\ - 1 \end{array} \right] + \mu \left[ \begin{array} { r } 1 \\ 0 \\ - 2 \end{array} \right]\).
   1. Show that the point \(P ( - 2 , - 3,5 )\) lies on \(l _ { 2 }\).
   2. The point \(Q\) lies on \(l _ { 1 }\) and is such that \(P Q\) is perpendicular to \(l _ { 2 }\). Find the coordinates of \(Q\).

8 The points \(A\) and \(B\) have coordinates \(( 2,1 , - 1 )\) and \(( 3,1 , - 2 )\) respectively. The angle \(O B A\) is \(\theta\), where \(O\) is the origin.

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Show that \(\cos \theta = \frac { 5 } { 2 \sqrt { 7 } }\).
2. The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O B }\). The line \(l\) is parallel to \(\overrightarrow { A B }\) and passes through the point \(C\). Find a vector equation of \(l\).
3. The point \(D\) lies on \(l\) such that angle \(O D C = 90 ^ { \circ }\). Find the coordinates of \(D\).

8 The points \(A , B\) and \(C\) have coordinates \(( 2 , - 1 , - 5 ) , ( 0,5 , - 9 )\) and \(( 9,2,3 )\) respectively. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 2 \\ - 1 \\ - 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]\).

1. Verify that the point \(B\) lies on the line \(l\).
2. Find the vector \(\overrightarrow { B C }\).
3. The point \(D\) is such that \(\overrightarrow { A D } = 2 \overrightarrow { B C }\).
   1. Show that \(D\) has coordinates \(( 20 , - 7,19 )\).
   2. The point \(P\) lies on \(l\) where \(\lambda = p\). The line \(P D\) is perpendicular to \(l\). Find the value of \(p\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \left[ \begin{array} { r } 8 \\ 6 \\ - 9 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ - 3 \\ - 1 \end{array} \right]\) and \(\mathbf { r } = \left[ \begin{array} { r } - 4 \\ 0 \\ 11 \end{array} \right] + \mu \left[ \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right]\) respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection, \(P\).
3. The point \(A ( - 4,0,11 )\) lies on \(l _ { 2 }\). The point \(B\) on \(l _ { 1 }\) is such that \(A P = B P\). Find the length of \(A B\).