Area of parallelogram or trapezium using vectors

10 \includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-16_318_1006_260_568} Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\), shown in the diagram, are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 1 \\ 3 \\ - 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 4 \\ - 2 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 2 \\ 2 \\ - 1 \end{array} \right) .$$

1. Show that \(A B\) is perpendicular to \(B C\).
2. Show that \(A B C D\) is a trapezium.
3. Find the area of \(A B C D\), giving your answer correct to 2 decimal places.

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.

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}

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 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 \(B\).