Area of triangle using cross product

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 \(A B C\).
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 \(a\).

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 \(ABC\). [4]
2. Show that \(\frac{1}{6}\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0\) [3]
3. Hence or otherwise, state what can be deduced about the vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\). [1]

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 \(\overrightarrow{AB} \times \overrightarrow{AC}\), prove that the area of \(\triangle ABC\) can be expressed in the form \(\frac{1}{2}|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|\). [5]

The points \(A\), \(B\), \(C\), and \(D\) have position vectors $$\mathbf{a} = 2\mathbf{i} + \mathbf{k}, \quad \mathbf{b} = \mathbf{i} + 3\mathbf{j}, \quad \mathbf{c} = \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}, \quad \mathbf{d} = 4\mathbf{j} + \mathbf{k}$$ respectively.

1. Find \(\overrightarrow{AB} \times \overrightarrow{AC}\) and hence find the area of triangle \(ABC\). [7]
2. Find the volume of the tetrahedron \(ABCD\). [2]
3. Find the perpendicular distance of \(D\) from the plane containing \(A\), \(B\) and \(C\). [3]

(Total 12 marks)

(In this question, the notation \(\Delta ABC\) denotes the area of the triangle \(ABC\).) The points \(P\), \(Q\) and \(R\) have position vectors \(p\mathbf{i}\), \(q\mathbf{j}\) and \(r\mathbf{k}\) respectively, relative to the origin \(O\), where \(p\), \(q\) and \(r\) are positive. The points \(O\), \(P\), \(Q\) and \(R\) are joined to form a tetrahedron.

1. Draw a sketch of the tetrahedron and write down the values of \(\Delta OPQ\), \(\Delta OQR\) and \(\Delta ORP\). [3]
2. Use the definition of the vector product to show that \(\frac{1}{2}|\overrightarrow{RP} \times \overrightarrow{RQ}| = \Delta PQR\). [1]
3. Show that \((\Delta OPQ)^2 + (\Delta OQR)^2 + (\Delta ORP)^2 = (\Delta PQR)^2\). [6]

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

1. Find the area of the triangle \(ABC\) [4 marks]
2. The points \(A\), \(B\) and \(C\) all lie in the plane \(\Pi\) Find an equation of the plane \(\Pi\), in the form \(\mathbf{r} \cdot \mathbf{n} = d\) [2 marks]
3. The point \(P\) has position vector \(\mathbf{p} = \mathbf{i} + 4\mathbf{j} + 2\mathbf{k}\) Find the exact distance of \(P\) from \(\Pi\) [3 marks]

The points \(A\), \(B\) and \(C\) have coordinates \(A(4, 5, 2)\), \(B(-3, 2, -4)\) and \(C(2, 6, 1)\)

1. Use a vector product to show that the area of triangle \(ABC\) is \(\frac{5\sqrt{11}}{2}\) [4 marks]
2. The points \(A\), \(B\) and \(C\) lie in a plane. Find a vector equation of the plane in the form \(\mathbf{r} \cdot \mathbf{n} = k\) [1 mark]
3. Hence find the exact distance of the plane from the origin. [1 mark]

Points \(A\), \(B\) and \(C\) have coordinates \((4, 2, 0)\), \((1, 5, 3)\) and \((1, 4, -2)\) respectively. The line \(l\) passes through \(A\) and \(B\).

1. Find a cartesian equation for \(l\). [3]

\(M\) is the point on \(l\) that is closest to \(C\).

1. Find the coordinates of \(M\). [4]
2. Find the exact area of the triangle \(ABC\). [4]