Volume of tetrahedron using scalar triple product - Vectors: Cross Product & Distances

OCR Further Additional Pure 2017 Specimen Q2

3 marks Moderate -0.5

Find the volume of tetrahedron OABC, where O is the origin, A = (2, 3, 1), B = (-4, 2, 5) and C = (1, 4, 4). [3]

Pre-U Pre-U 9795/1 2011 June Q9

11 marks Standard +0.3

The points $A$, $B$ and $C$ have position vectors $$\mathbf{a} = \begin{pmatrix} 19 \\ 3 \\ 10 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 12 \\ 7 \\ -1 \end{pmatrix} \quad \text{and} \quad \mathbf{c} = \begin{pmatrix} 5 \\ 15 \\ 3 \end{pmatrix}$$ respectively, and $O$ is the origin. Calculate the volume of the tetrahedron $OABC$. [3]
1. The plane $\Pi_1$ has equation $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 6 \\ 2 \\ 5 \end{pmatrix}$. Determine an equation for $\Pi_1$ in the form $\mathbf{r} \cdot \mathbf{n} = d$. [4]
2. A second plane, $\Pi_2$, has equation $\mathbf{r} \cdot \begin{pmatrix} 1 \\ 4 \\ 7 \end{pmatrix} = 13$. Find a vector equation for the line of intersection of $\Pi_1$ and $\Pi_2$. [4]

Pre-U Pre-U 9795/1 2013 November Q7

8 marks Standard +0.3

Relative to an origin $O$, the points $P$, $Q$ and $R$ have position vectors $$\mathbf{p} = \mathbf{i} + 2\mathbf{j} - 7\mathbf{k}, \quad \mathbf{q} = -3\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{r} = 6\mathbf{i} + 4\mathbf{j} + \alpha\mathbf{k}$$ respectively.

Determine $\mathbf{p} \times \mathbf{q}$. [2]
Deduce the value of $\alpha$ for which
1. $OR$ is normal to the plane $OPQ$, [1]
2. the volume of tetrahedron $OPQR$ is 50, [3]
3. $R$ lies in the plane $OPQ$. [2]

Pre-U Pre-U 9795/1 2015 June Q1

3 marks Moderate -0.5

Determine the volume of tetrahedron $OABC$, where $O$ is the origin and $A$, $B$ and $C$ are, respectively, the points $(2, 3, -2)$, $(2, 0, 4)$ and $(6, 1, 7)$. [3]