Area of triangle using cross product

8. The position vectors of the points \(A , B\) and \(C\) from a fixed origin \(O\) are $$\mathbf { a } = \mathbf { i } - \mathbf { j } , \quad \mathbf { b } = \mathbf { i } + \mathbf { j } + \mathbf { k } , \quad \mathbf { c } = 2 \mathbf { j } + \mathbf { k }$$ respectively.

1. Using vector products, find the area of the triangle \(A B C\).
2. Show that \(\frac { 1 } { 6 } \mathbf { a } . ( \mathbf { b } \times \mathbf { c } ) = 0\)
3. Hence or otherwise, state what can be deduced about the vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).

6 The points \(A , B\) and \(C\) have position vectors \(2 \mathbf { i } - \mathbf { j } + \mathbf { k } , 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) and \(- \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) respectively.

1. Find the area of the triangle \(A B C\).
   .................................................................................................................................... \includegraphics[max width=\textwidth, alt={}]{9221f480-4af6-44be-a535-d2ceb0f8b5d2-08_72_1566_484_328} ................................................................................................................................... .................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{9221f480-4af6-44be-a535-d2ceb0f8b5d2-08_71_1563_772_331} \includegraphics[max width=\textwidth, alt={}, center]{9221f480-4af6-44be-a535-d2ceb0f8b5d2-08_71_1563_868_331}
2. Find the perpendicular distance of the point \(A\) from the line \(B C\).
3. Find the cartesian equation of the plane through \(A , B\) and \(C\).

6 The points \(A , B\) and \(C\) have position vectors \(2 \mathbf { i } - \mathbf { j } + \mathbf { k } , 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) and \(- \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) respectively.

1. Find the area of the triangle \(A B C\).
   .................................................................................................................................... \includegraphics[max width=\textwidth, alt={}]{a0987277-06e9-451b-ae18-bb7de9e7661c-08_72_1566_484_328} .................................................................................................................................... .................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a0987277-06e9-451b-ae18-bb7de9e7661c-08_71_1563_772_331} \includegraphics[max width=\textwidth, alt={}, center]{a0987277-06e9-451b-ae18-bb7de9e7661c-08_71_1563_868_331}
2. Find the perpendicular distance of the point \(A\) from the line \(B C\).
3. Find the cartesian equation of the plane through \(A , B\) and \(C\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a parallelogram \(X A P B\).
Given that \(\overrightarrow { O X } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right)\) $$\begin{aligned} & \overrightarrow { O A } = \left( \begin{array} { l } 0 \\ 4 \\ 1 \end{array} \right) \\ & \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 3 \\ 1 \end{array} \right) \end{aligned}$$ a. Find the coordinates of the point \(P\).
b. Show that \(X A P B\) is a rhombus.
c. Find the exact area of the rhombus \(X A P B\).

12 Three intersecting lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have equations \(L _ { 1 } : \frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 1 } , \quad L _ { 2 } : \frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { - 4 } \quad\) and \(\quad L _ { 3 } : \frac { x - 1 } { 1 } = \frac { y - 2 } { 1 } = \frac { z + 4 } { 5 }\).
Find the area of the triangle enclosed by these lines.

1. The points \(A , B\) and \(C\) are the vertices of a triangle.

Given that

• \(\overrightarrow { A B } = \left( \begin{array} { l } p \\ 4 \\ 6 \end{array} \right)\) and \(\overrightarrow { A C } = \left( \begin{array} { l } q \\ 4 \\ 5 \end{array} \right)\) where \(p\) and \(q\) are constants
• \(\overrightarrow { A B } \times \overrightarrow { A C }\) is parallel to \(2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\)
  1. determine the value of \(p\) and the value of \(q\)
  2. Hence, determine the exact area of triangle \(A B C\)

1. Vectors \(\mathbf { u }\) and \(\mathbf { v }\) are given by

$$\mathbf { u } = 5 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \mathbf { v } = a \mathbf { i } - 6 \mathbf { j } + 2 \mathbf { k }$$ where \(a\) is a constant.

1. Determine, in terms of \(a\), the vector product \(\mathbf { u } \times \mathbf { v }\) Given that
   • \(\overrightarrow { A B } = 2 \mathbf { u }\)
   • \(\overrightarrow { A C } = \mathbf { v }\)
   • the area of triangle \(A B C\) is 15
   • determine the possible values of \(a\).

17. Referred to a fixed origin \(O\), the position vectors of three non-collinear points \(A , B\) and \(C\) are \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively. By considering \(\overrightarrow { A B } \times \overrightarrow { A C }\), prove that the area of \(\triangle A B C\) can be expressed in the form \(\frac { 1 } { 2 } | \mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a } |\).
[0pt] [P6 June 2003 Qn 1]

8 The vectors \(\mathbf { a } , \mathbf { b }\), and \(\mathbf { c }\) are such that \(\mathbf { a } \times \mathbf { b } = \left[ \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right]\) and \(\mathbf { a } \times \mathbf { c } = \left[ \begin{array} { l } 0 \\ 0 \\ 3 \end{array} \right]\) Work out \(( \mathbf { a } - \mathbf { 4 } \mathbf { b } + \mathbf { 3 c } ) \times ( \mathbf { 2 a } )\) [0pt] [4 marks]

1

1. Find a vector which is perpendicular to both \(\left( \begin{array} { r } 1 \\ 3 \\ - 2 \end{array} \right)\) and \(\left( \begin{array} { r } - 3 \\ - 6 \\ 4 \end{array} \right)\).
2. The cartesian equation of a line is \(\frac { x } { 2 } = y - 3 = 2 z + 4\). Express the equation of this line in vector form.