Further Mathematics · OCR H245 · Option A · Topic 8.04

Further vectors

Vector product.

1 sub-topic
Taught in Year 12
17 practice questions
1 animated lesson

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What the specification says

Every sub-topic OCR can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

8.04a-eVector productY12

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Further vectors

Easy

Volumes from the scalar triple product · Year 12 · 1 mark

Three edges of a tetrahedron that meet at one vertex are given by the vectors $a$ , $b$ and $c$ . What is its volume?
(1)

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A tetrahedron has one sixth of the volume of the parallelepiped with the same three edges. $C$ : $\tfrac{1}{6}|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|$

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further vectors

Medium

Scalar triple product · Year 12 · 2 marks

Find the volume of the parallelepiped with edges $-2\mathbf{j}$ , $\mathbf{i}$ ${}+ 2\mathbf{j}$ ${}+ 3\mathbf{k}$ , $-2\mathbf{i}$ ${}+ 2\mathbf{j}$ ${}+ \mathbf{k}$ .
(2)

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$|\mathbf{a}\cdot(\mathbf{b} \times \mathbf{c})|$
$14$

Total2

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Further vectors

Hard

Volume of a tetrahedron · Year 12 · 5 marks

The points

A

B

and

C

have position vectors

\begin{pmatrix} -2 \\ 1 \\ 1 \end{pmatrix}

\begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}

and

\begin{pmatrix} 1 \\ -2 \\ k \end{pmatrix}

relative to the origin

O

, where

k

is a constant. The tetrahedron OABC has volume 3.

Find the possible values of $k$ .
(5)

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The volume of a tetrahedron. $V$ ${}= \tfrac{1}{6}|\mathbf{a}\cdot(\mathbf{b} \times \mathbf{c})|$
$\mathbf{b} \times \mathbf{c}$ ${}= \begin{pmatrix} 2 - 2 k \\ 1 - 2 k \\ -2 \end{pmatrix}$
The scalar triple product, then the modulus. $|2 k - 5| = 18$
$k$ ${}= - \frac{13}{2} \text{ or } \frac{23}{2}$

Total5

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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