1.10c Magnitude and direction: of vectors

1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter.

The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 0 \\ 9 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right)\), where \(\mu\) is a scalar parameter.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(C\), find

1. the coordinates of \(C\). The point \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 0\) and the point \(B\) is the point on \(l _ { 2 }\) where \(\mu = - 1\)
2. Find the size of the angle \(A C B\). Give your answer in degrees to 2 decimal places.
3. Hence, or otherwise, find the area of the triangle \(A B C\).

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of parallelogram \(A B C D\).
Given that \(\overrightarrow { A B } = 6 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { B C } = 2 \mathbf { i } + 5 \mathbf { j } + 8 \mathbf { k }\)

1. find the size of angle \(A B C\), giving your answer in degrees, to 2 decimal places.
2. Find the area of parallelogram \(A B C D\), giving your answer to one decimal place.

5. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ 4 \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 3 \\ 6 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 13 \\ - 1 \\ 4 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ 1 \\ - 3 \end{array} \right)$$ 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 \(A\).
2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to one decimal place. A circle with centre \(A\) and radius 35 cuts the line \(l _ { 1 }\) at the points \(P\) and \(Q\). Given that the \(x\) coordinate of \(P\) is greater than the \(x\) coordinate of \(Q\),
3. find the coordinates of \(P\) and the coordinates of \(Q\). \section*{Question 5 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 5 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 5 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

1. 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 \(8 \mathbf { i } + 3 \mathbf { j } - 7 \mathbf { k }\)

The line \(l\) passes through \(A\) and \(B\).

1. 1. Find \(\overrightarrow { A B }\)
   2. Find a vector equation for the line \(l\) The point \(C\) has position vector \(3 \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k }\) The point \(P\) lies on \(l\) Given that \(\overrightarrow { C P }\) is perpendicular to \(l\)
2. find the position vector of the point \(P\)

1. Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations

$$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 3 \mathbf { i } + p \mathbf { j } + 7 \mathbf { k } ) + \lambda ( 2 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 8 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } ) + \mu ( 4 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect,

1. find the value of \(p\),
2. find the position vector of the point of intersection.
3. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) Give your answer in degrees to one decimal place. The point \(A\) lies on \(l _ { 1 }\) with parameter \(\lambda = 2\) The point \(B\) lies on \(l _ { 2 }\) with \(\overrightarrow { A B }\) perpendicular to \(l _ { 2 }\)
4. Find the coordinates of \(B\)

7. The point \(A\) has position vector \(\mathbf { a } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and the point \(B\) has position vector \(\mathbf { b } = \mathbf { i } + \mathbf { j } - 4 \mathbf { k }\), relative to an origin \(O\).

1. Find the position vector of the point \(C\), with position vector \(\mathbf { c }\), given by $$\mathbf { c } = \mathbf { a } + \mathbf { b } .$$
2. Show that \(O A C B\) is a rectangle, and find its exact area. The diagonals of the rectangle, \(A B\) and \(O C\), meet at the point \(D\).
3. Write down the position vector of the point \(D\).
4. Find the size of the angle \(A D C\).

7. Relative to a fixed origin \(O\), the point \(A\) has position vector ( \(2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }\) ), the point \(B\) has position vector \(( 5 \mathbf { i } + 2 \mathbf { j } + 10 \mathbf { k } )\), and the point \(D\) has position vector \(( - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\). The line \(l\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l\).
3. Show that the size of the angle \(B A D\) is \(109 ^ { \circ }\), to the nearest degree. The points \(A , B\) and \(D\), together with a point \(C\), are the vertices of the parallelogram \(A B C D\), where \(\overrightarrow { A B } = \overrightarrow { D C }\).
4. Find the position vector of \(C\).
5. Find the area of the parallelogram \(A B C D\), giving your answer to 3 significant figures.
6. Find the shortest distance from the point \(D\) to the line \(l\), giving your answer to 3 significant figures.

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}

5. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 0 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right)\). The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 3 \\ 6 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)\).

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(A\) is on \(l _ { 1 }\) where \(\lambda = 1\), and the point \(B\) is on \(l _ { 2 }\) where \(\mu = 2\).
2. Find the cosine of the acute angle between \(A B\) and \(l _ { 1 }\).

7. Relative to a fixed origin \(O\), the point \(A\) has position vector \(( 8 \mathbf { i } + 13 \mathbf { j } - 2 \mathbf { k } )\), the point \(B\) has position vector ( \(10 \mathbf { i } + 14 \mathbf { j } - 4 \mathbf { k }\) ), and the point \(C\) has position vector \(( 9 \mathbf { i } + 9 \mathbf { j } + 6 \mathbf { k } )\). The line \(l\) passes through the points \(A\) and \(B\).

1. Find a vector equation for the line \(l\).
2. Find \(| \overrightarrow { C B } |\).
3. Find the size of the acute angle between the line segment \(C B\) and the line \(l\), giving your answer in degrees to 1 decimal place.
4. Find the shortest distance from the point \(C\) to the line \(l\). The point \(X\) lies on \(l\). Given that the vector \(\overrightarrow { C X }\) is perpendicular to \(l\),
5. find the area of the triangle \(C X B\), giving your answer to 3 significant figures.

6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 6 \\ - 3 \\ - 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 5 \\ 15 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right)$$ 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 \(A\).
2. Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). The point \(B\) has position vector \(\left( \begin{array} { r } 5 \\ - 1 \\ 1 \end{array} \right)\).
3. Show that \(B\) lies on \(l _ { 1 }\).
4. Find the shortest distance from \(B\) to the line \(l _ { 2 }\), giving your answer to 3 significant figures. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued}

1. Relative to a fixed origin \(O\), the point \(A\) has position vector \(( 10 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\), and the point \(B\) has position vector \(( 8 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } )\).

The line \(l\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l\). The point \(C\) has position vector \(( 3 \mathbf { i } + 12 \mathbf { j } + 3 \mathbf { k } )\).
   The point \(P\) lies on \(l\). Given that the vector \(\overrightarrow { C P }\) is perpendicular to \(l\),
3. find the position vector of the point \(P\).

6. Relative to a fixed origin \(O\), the point \(A\) has position vector \(21 \mathbf { i } - 17 \mathbf { j } + 6 \mathbf { k }\) and the point \(B\) has position vector \(25 \mathbf { i } - 14 \mathbf { j } + 18 \mathbf { k }\). The line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } a \\ b \\ 10 \end{array} \right) + \lambda \left( \begin{array} { r } 6 \\ c \\ - 1 \end{array} \right)$$ where \(a , b\) and \(c\) are constants and \(\lambda\) is a parameter.
Given that the point \(A\) lies on the line \(l\),

1. find the value of \(a\). Given also that the vector \(\overrightarrow { A B }\) is perpendicular to \(l\),
2. find the values of \(b\) and \(c\),
3. find the distance \(A B\). The image of the point \(B\) after reflection in the line \(l\) is the point \(B ^ { \prime }\).
4. Find the position vector of the point \(B ^ { \prime }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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 \(B\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) Question 8 continued

6. With respect to a fixed origin, the point \(A\) with position vector \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) lies on the line \(l _ { 1 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 2 \\ - 1 \end{array} \right) , \quad \text { where } \lambda \text { is a scalar parameter, }$$ and the point \(B\) with position vector \(4 \mathbf { i } + p \mathbf { j } + 3 \mathbf { k }\), where \(p\) is a constant, lies on the line \(l _ { 2 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 7 \\ 0 \\ 7 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 5 \\ 4 \end{array} \right) , \quad \text { where } \mu \text { is a scalar parameter. }$$

1. Find the value of the constant \(p\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the position vector of their point of intersection, \(C\).
3. Find the size of the angle \(A C B\), giving your answer in degrees to 3 significant figures.
4. Find the area of the triangle \(A B C\), giving your answer to 3 significant figures. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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. The point \(A\) with coordinates ( \(- 3,7,2\) ) lies on a line \(l _ { 1 }\) The point \(B\) also lies on the line \(l _ { 1 }\) Given that \(\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)\),

1. find the coordinates of point \(B\). The point \(P\) has coordinates ( \(9,1,8\) )
2. Find the cosine of the angle \(P A B\), giving your answer as a simplified surd.
3. Find the exact area of triangle \(P A B\), giving your answer in its simplest form. The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
4. Find a vector equation for the line \(l _ { 2 }\) The point \(Q\) lies on the line \(l _ { 2 }\) Given that the line segment \(A P\) is perpendicular to the line segment \(B Q\),
5. find the coordinates of the point \(Q\).

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

$$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 10 \\ - 9 \end{array} \right) + \lambda \left( \begin{array} { l } 4 \\ 4 \\ 2 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter }$$ Given that \(\overrightarrow { O A }\) is a unit vector parallel to \(l\),

1. find \(\overrightarrow { O A }\) The point \(X\) lies on \(l\).
   Given that \(X\) is the point on \(l\) that is closest to the origin,
2. find the coordinates of \(X\). The points \(O , X\) and \(A\) form the triangle \(O X A\).
3. Find the exact area of triangle \(O X A\).

1. Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(\quad \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\)
• the point \(C\) has position vector \(3 \mathbf { i } + p \mathbf { j } - \mathbf { k }\) where \(p\) is a constant.
  The line \(l\) passes through \(A\) and \(B\).
  1. Find a vector equation for the line \(l\)

Given that \(\overrightarrow { A C }\) is perpendicular to \(l\)

find the value of \(p\)

Hence find the area of triangle \(A B C\), giving your answer as a surd in simplest form.

1. Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } + \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\)
• the point \(P\) has position vector \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\)

The straight line \(l\) passes through \(A\) and \(B\).

1. Find a vector equation for \(l\). The point \(C\) lies on \(l\) so that \(P C\) is perpendicular to \(l\).
2. Find the coordinates of \(C\). The point \(P ^ { \prime }\) is the reflection of \(P\) in the line \(l\).
3. Find the coordinates of \(P ^ { \prime }\)
4. Hence find \(\left| \overrightarrow { P P ^ { \prime } } \right|\), giving your answer as a simplified surd.

1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 7 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 2 \end{array} \right)\) where \(\lambda\) is a scalar parameter.

The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { r } 4 \\ - 1 \\ 8 \end{array} \right)\) where \(\mu\) is a scalar parameter.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\)

1. state the coordinates of \(P\) Given that the angle between lines \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\)
2. find the value of \(\cos \theta\), giving the answer as a fully simplified fraction. The point \(Q\) lies on \(l _ { 1 }\) where \(\lambda = 6\) Given that point \(R\) lies on \(l _ { 2 }\) such that triangle \(Q P R\) is an isosceles triangle with \(P Q = P R\)
3. find the exact area of triangle \(Q P R\)
4. find the coordinates of the possible positions of point \(R\)

1. 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 found. 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 \(E\).

7. The plane \(\Pi _ { 1 }\) has equation \(x + y + z = 3\) and the plane \(\Pi _ { 2 }\) has equation \(2 x + 3 y - z = 4\) The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(L\).

1. Find a cartesian equation for the line \(L\). The plane \(\Pi _ { 3 }\) has equation $$\text { r. } \left( \begin{array} { r } 5 \\ - 4 \\ 4 \end{array} \right) = 12$$ The line \(L\) meets the plane \(\Pi _ { 3 }\) at the point \(A\).
2. Find the coordinates of \(A\).
3. Find the acute angle between \(\overrightarrow { O A }\) and the line \(L\), where \(O\) is the origin. Give your answer in degrees to one decimal place.

1. A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. The velocity of \(P\) is \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 0\), and \(( 7 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 5 \mathrm {~s}\).

Find

1. the speed of \(P\) at \(t = 0\),
2. the vector \(\mathbf { F }\) in the form \(a \mathbf { i } + b \mathbf { j }\),
3. the value of \(t\) when \(P\) is moving parallel to \(\mathbf { i }\).