Further Mathematics · Pearson Edexcel 9FM0 · Option FP1 · Topic 5

Further vectors

The vector product, the scalar triple product for volumes, and writing lines and planes using the vector product.

  • 3 sub-topics
  • 2 in Year 12, 1 in Year 13
  • 46 practice questions
  • 3 animated lessons
Practice questions (46) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel 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.

  1. 5.1Vector productY12
  2. 5.2Scalar triple productY12
  3. 5.3Vector geometry with the vector productY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 46 questions in this topic, free, no account needed.

Further vectors

Easy

Vector product · Year 12 · 2 marks

a\mathbf{a}=3j−k{}= 3\mathbf{j} - \mathbf{k}, b\mathbf{b}=−2i{}= -2\mathbf{i}+3j{}+ 3\mathbf{j}+2k{}+ 2\mathbf{k}.
  1. Find a×ba \times b.
    (2)
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  1. Use the determinant with i\mathrm{i}, jj, kk in the first row.
  2. 9i9\mathbf{i}+2j{}+ 2\mathbf{j}+6k{}+ 6\mathbf{k}

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Further vectors

Medium

The vector product form of a line · Year 12 · 5 marks

The line ll passes through the point with position vector (−33−1)\begin{pmatrix} -3 \\ 3 \\ -1 \end{pmatrix} and is parallel to (3−33)\begin{pmatrix} 3 \\ -3 \\ 3 \end{pmatrix}.
  1. (a)
    Find an equation of ll in the form r×d\mathbf{r} \times \mathbf{d}=m{}= \mathbf{m}.
    (3)
  2. (b)
    Determine whether the point with position vector (−430)\begin{pmatrix} -4 \\ 3 \\ 0 \end{pmatrix} lies on ll.
    (2)
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Part (a)3 marks

  1. r−ar - a is parallel to dd.(r−a)×d(\mathbf{r} - \mathbf{a}) \times \mathbf{d}=0{}= \mathbf{0}⇒r×d{}\Rightarrow \mathbf{r} \times \mathbf{d}=a×d{}= \mathbf{a} \times \mathbf{d}
  2. a×d\mathbf{a} \times \mathbf{d}=|ijk−33−13−33|{}= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -3 & 3 & -1 \\ 3 & -3 & 3 \end{vmatrix}
  3. r×(3−33)\mathbf{r} \times \begin{pmatrix} 3 \\ -3 \\ 3 \end{pmatrix}=(660){}= \begin{pmatrix} 6 \\ 6 \\ 0 \end{pmatrix}

Part (b)2 marks

  1. (−430)×(3−33)\begin{pmatrix} -4 \\ 3 \\ 0 \end{pmatrix} \times \begin{pmatrix} 3 \\ -3 \\ 3 \end{pmatrix}=(9123){}= \begin{pmatrix} 9 \\ 12 \\ 3 \end{pmatrix}
  2. Not equal to mm, so the point does not lie on ll.

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Further vectors

Hard

Volume of a tetrahedron · Year 12 · 5 marks

The points AA, BB and CC have position vectors (−211)\begin{pmatrix} -2 \\ 1 \\ 1 \end{pmatrix}, (2−21)\begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix} and (1−2k)\begin{pmatrix} 1 \\ -2 \\ k \end{pmatrix} relative to the origin OO, where kk is a constant. The tetrahedron OABC has volume 3.
  1. Find the possible values of kk.
    (5)
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  1. The volume of a tetrahedron.VV=16∣a⋅(b×c)∣{}= \tfrac{1}{6}|\mathbf{a}\cdot(\mathbf{b} \times \mathbf{c})|
  2. b×c\mathbf{b} \times \mathbf{c}=(2−2k1−2k−2){}= \begin{pmatrix} 2 - 2 k \\ 1 - 2 k \\ -2 \end{pmatrix}
  3. The scalar triple product, then the modulus.∣2k−5∣=18|2 k - 5| = 18
  4. kk=−132 or 232{}= - \frac{13}{2} \text{ or } \frac{23}{2}

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