1.10b Vectors in 3D: i,j,k notation

7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 13 \mathbf { i } + 15 \mathbf { j } - 8 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 7 \mathbf { i } - 6 \mathbf { j } + 14 \mathbf { k } ) + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ 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, \(B\).
2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The point \(A\) has position vector \(- 5 \mathbf { i } - 3 \mathbf { j } + 16 \mathbf { k }\)
3. Show that \(A\) lies on \(l _ { 1 }\) The point \(C\) lies on the line \(l _ { 1 }\) where \(\overrightarrow { A B } = \overrightarrow { B C }\)
4. Find the position vector of \(C\).
   
   \section*{"}

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.

14. Relative to a fixed origin \(O\), the line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 4 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. Points \(A\) and \(B\) lie on the line \(l\), where \(A\) has coordinates ( \(1 , a , 5\) ) and \(B\) has coordinates ( \(b , - 1,3\) ).

1. Find the value of the constant \(a\) and the value of the constant \(b\).
2. Find the vector \(\overrightarrow { A B }\). The point \(C\) has coordinates ( \(4 , - 3,2\) )
3. Show that the size of the angle \(C A B\) is \(30 ^ { \circ }\)
4. Find the exact area of the triangle \(C A B\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined. The point \(D\) lies on the line \(l\) so that the area of the triangle \(C A D\) is twice the area of the triangle \(C A B\).
5. Find the coordinates of the two possible positions of \(D\).

1. (i) Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } - 5 \\ 1 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right) \text { where } \lambda \text { is a scalar parameter. }$$ The point \(P\) lies on \(l _ { 1 }\). Given that \(\overrightarrow { O P }\) is perpendicular to \(l _ { 1 }\), calculate the coordinates of \(P\).
(ii) Relative to a fixed origin \(O\), the line \(l _ { 2 }\) is given by the equation $$l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ - 3 \\ 12 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 3 \\ 4 \end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$ The point \(A\) does not lie on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O A }\) is parallel to the line \(l _ { 2 }\) and \(| \overrightarrow { O A } | = \sqrt { 2 }\) units, calculate the possible position vectors of the point \(A\).

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

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(P\) is on \(l _ { 1 }\) where \(\lambda = 0\), and the point \(Q\) is on \(l _ { 2 }\) where \(\mu = - 1\)
2. Find the acute angle between the line segment \(P Q\) and \(l _ { 1 }\), giving your answer in degrees to 2 decimal places.
3. Find the shortest distance from the point \(Q\) to the line \(l _ { 1 }\), giving your answer to 3 significant figures.

1. Relative to a fixed origin \(O\),
   the point \(A\) has position vector \(( 2 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\) the point \(B\) has position vector \(( 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\) the point \(C\) has position vector ( \(2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\) )

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 \(C A B\) is \(62.8 ^ { \circ }\), to one decimal place.
4. Hence find the area of triangle \(C A B\), giving your answer to 3 significant figures. The point \(D\) lies on the line \(l\). Given that the area of triangle \(C A D\) is twice the area of triangle \(C A B\),
5. find the two possible position vectors of point \(D\).
   

11. Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 4 \\ 3 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) passes through the origin and is parallel to \(l _ { 1 }\)

1. Find a vector equation for \(l _ { 2 }\) The point \(A\) and the point \(B\) both lie on \(l _ { 1 }\) with parameters \(\lambda = 0\) and \(\lambda = 3\) respectively.
   Write down
2. 1. the coordinates of \(A\),
   2. the coordinates of \(B\).
3. Find the size of the acute angle between \(O A\) and \(l _ { 1 }\) Give your answer in degrees to one decimal place. The point \(D\) lies on \(l _ { 2 }\) such that \(O A B D\) is a parallelogram.
4. Find the area of \(O A B D\), giving your answer to the nearest whole number.

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} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

6. The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\) respectively. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l _ { 1 }\). A second line \(l _ { 2 }\) passes through the origin and is parallel to the vector \(\mathbf { i } + \mathbf { k }\). The line \(l _ { 1 }\) meets the line \(l _ { 2 }\) at the point \(C\).
3. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
4. Find the position vector of the point \(C\).

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.

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.

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 \(B\).

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.

5. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\).

1. Show that the magnitude of the acceleration of \(P\) is \(2 \sqrt { 13 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the velocity of \(P\) at time \(t = 2\) seconds. Another particle \(Q\) moves with constant velocity \(\mathbf { v } = ( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
3. Find the distance moved by \(Q\) in 2 seconds.
4. Show that at time \(t = 3.5\) seconds both particles are moving in the same direction.

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]

Two cars \(P\) and \(Q\) are moving on straight horizontal roads with constant velocities. The velocity of \(P\) is \(( 15 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( 20 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the direction of motion of \(Q\), giving your answer as a bearing to the nearest degree. At time \(t = 0\), the position vector of \(P\) is \(400 \mathbf { i }\) metres and the position vector of \(Q\) is 800j metres. At time \(t\) seconds, the position vectors of \(P\) and \(Q\) are \(\mathbf { p }\) metres and \(\mathbf { q }\) metres respectively.
2. Find an expression for
   1. \(\mathbf { p }\) in terms of \(t\),
   2. \(\mathbf { q }\) in terms of \(t\).
3. Find the position vector of \(Q\) when \(Q\) is due west of \(P\).

5. A t time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 4 \right) \mathbf { i } + ( 2 t - 4 ) \mathbf { j }$$ When \(t = 0 , P\) is at the fixed point \(O\).

1. Find the acceleration of \(P\) at the instant when \(t = 0\)
2. Find the exact speed of \(P\) at the instant when \(P\) is moving in the direction of the vector \(( 11 \mathbf { i } + \mathbf { j } )\) for the second time.
3. Show that \(P\) never returns to \(O\). \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-14_2658_1938_107_123} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-15_149_140_2604_1818}

5. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = \left( 5 t ^ { 2 } - 12 t + 15 \right) \mathbf { i } + \left( t ^ { 2 } + 8 t - 10 \right) \mathbf { j }$$ When \(t = 0 , P\) is at the origin \(O\).
At time \(T\) seconds, \(P\) is moving in the direction of \(( \mathbf { i } + \mathbf { j } )\).

1. Find the value of \(T\). When \(t = 3 , P\) is at the point \(A\).
2. Find the magnitude of the acceleration of \(P\) as it passes through \(A\).
3. Find the position vector of \(A\).