Area of triangle from given side vectors or coordinates

6 Relative to an origin \(O\), the position vector of \(A\) is \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and the position vector of \(B\) is \(7 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\).

1. Show that angle \(O A B\) is a right angle.
2. Find the area of triangle \(O A B\).

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

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 7 \end{array} \right) + \mu \left( \begin{array} { l } 8 \\ 4 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).

1. Write down the coordinates of \(A\). Given that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\),
2. show that \(\sin \theta = k \sqrt { 2 }\), where \(k\) is a rational number to be found. The point \(B\) lies on \(l _ { 1 }\) where \(\lambda = 4\) The point \(C\) lies on \(l _ { 2 }\) such that \(A C = 2 A B\).
3. Find the exact area of triangle \(A B C\).
4. Find the coordinates of the two possible positions of \(C\).
   

4. Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { c } 6 \\ 3 \\ - 6 \end{array} \right)\) respectively. Find, in exact, simplified form,

1. the cosine of \(\angle A O B\),
2. the area of triangle \(O A B\),
3. the shortest distance from \(A\) to the line \(O B\).

5 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

3 A triangle ABC has vertices \(\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )\) and \(\mathrm { C } ( 4,8,3 )\). By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle. [6]

7 Points \(A ( 2,2,5 ) , B ( 1 , - 1 , - 4 ) , C ( 3,3,10 )\) and \(D ( 8,6,3 )\) are the vertices of a pyramid with a triangular base.

1. Calculate the lengths \(A B\) and \(A C\), and the angle \(B A C\).
2. Show that \(\overrightarrow { A D }\) is perpendicular to both \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\).
3. Calculate the volume of the pyramid \(A B C D\).
   [0pt] [The volume of the pyramid is \(V = \frac { 1 } { 3 } \times\) base area × perpendicular height.]

2 A triangle has vertices at \(A ( 1,1,3 ) , B ( 5,9 , - 5 )\) and \(C ( 6,5 , - 4 ) . P\) is the point on \(A B\) such that \(A P : P B = 3 : 1\).

1. Show that \(\overrightarrow { C P }\) is perpendicular to \(\overrightarrow { A B }\).
2. Find the area of the triangle \(A B C\).

4 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

9. The equations of the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) , \\ l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where \(\lambda\) and \(\mu\) are parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of \(Q\), their point of intersection.
2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\). The point \(P\) with \(x\)-coordinate 3 lies on the line \(l _ { 1 }\) and the point \(R\) with \(x\)-coordinate 4 lies on the line \(l _ { 2 }\).
3. Find, in its simplest form, the exact area of the triangle \(P Q R\). END

3 A triangle ABC has vertices \(\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )\) and \(\mathrm { C } ( 4,8,3 )\). By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle.

Relative to a fixed origin \(O\), the point \(A\) has position vector \(3\mathbf{i} + 2\mathbf{j} - \mathbf{k}\), the point \(B\) has position vector \(5\mathbf{i} + \mathbf{j} + \mathbf{k}\), and the point \(C\) has position vector \(7\mathbf{i} - \mathbf{j}\).

1. Find the cosine of angle \(ABC\). [4]
2. Find the exact value of the area of triangle \(ABC\). [3]

The point \(D\) has position vector \(7\mathbf{i} + 3\mathbf{k}\).

1. Show that \(AC\) is perpendicular to \(CD\). [2]
2. Find the ratio \(AD:DB\). [2]

The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the angle between \(AB\) and \(AC\). [6]
2. Find the area of triangle \(ABC\). [2]

Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} 1 \\ 5 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix}\) respectively. Find, in exact, simplified form,

1. the cosine of \(\angle AOB\), [4]
2. the area of triangle \(OAB\), [4]
3. the shortest distance from \(A\) to the line \(OB\). [2]

The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the angle between \(AB\) and \(AC\). [6]
2. Find the area of triangle \(ABC\). [2]