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Date	November 2016	Marks available	3	Reference code	16N.1.SL.TZ0.S_4
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find	Question number	S_4	Adapted from	N/A

Question

The position vectors of points P and Q are i $+$ 2 j $-$ k and 7i $+$ 3j $-$ 4k respectively.

Find a vector equation of the line that passes through P and Q.

[4]

a.

The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$ .

[3]

b.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

valid attempt to find direction vector (M1)

eg $\,\,\,\,\,$ $\overrightarrow {{\text{PQ}}} ,{\text{ }}\overrightarrow {{\text{QP}}}$

correct direction vector (or multiple of) (A1)

eg $\,\,\,\,\,$ 6i $+$ j $-$ 3k

any correct equation in the form r $=$ a $+$ tb (any parameter for $t$ ) A2 N3

where a is i $+$ 2j $-$ k or 7i $+$ 3j $-$ 4k , and b is a scalar multiple of 6i $+$ j $-$ 3k

eg $\,\,\,\,\,$ r $=$ 7i $+$ 3j $-$ 4k $+$ t(6i $+$ j $-$ 3k), r $= \left( {\begin{array}{*{20}{c}} {1 + 6s} \\ {2 + 1s} \\ { - 1 - 3s} \end{array}} \right),{\text{ }}r = \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} { - 6} \\ { - 1} \\ 3 \end{array}} \right)$

Notes: Award A1 for the form a $+$ tb, A1 for the form L $=$ a $+$ tb, A0 for the form r $=$ b $+$ ta.

[4 marks]

a.

correct expression for scalar product (A1)

eg $\,\,\,\,\,$ $6 \times 2 + 1 \times 0 + ( - 3) \times n,{\text{ }} - 3n + 12$

setting scalar product equal to zero (seen anywhere) (M1)

eg $\,\,\,\,\,$ u $\bullet$ v $= 0,{\text{ }} - 3n + 12 = 0$

$n = 4$ A1 N2

[3 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.13—Scalar (dot) product

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$\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}}$ .
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22M.2.AHL.TZ2.11e:
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17M.1.SL.TZ1.S_8a.ii:
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EXN.2.AHL.TZ0.11d.ii:
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19M.2.SL.TZ2.S_7:
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19M.1.SL.TZ2.S_2b:
perpendicular.
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18M.1.AHL.TZ2.H_1:
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Find the value of $y$ .
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Find $\mathop {{\text{PQ}}}\limits^ \to$ .
17N.1.SL.TZ0.S_9c:
The point D has coordinates $({q^2},{\text{ }}0,{\text{ }}q)$ . Given that $\overrightarrow {{\text{DC}}}$ is perpendicular to $L$ , find the possible values of $q$ .
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21N.2.AHL.TZ0.11c.ii:
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18M.1.SL.TZ2.S_1b:
The vector $\left( \begin{gathered} 2 \hfill \\ p \hfill \\ 0 \hfill \\ \end{gathered} \right)$ is perpendicular to $\mathop {{\text{AB}}}\limits^ \to$ . Find the value of p.
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Show that the Cartesian equation of the plane $Π$ that contains the triangle $ABC$ is $- x + y + z = - 2$ .
EXN.2.AHL.TZ0.11d.i:
Find a vector equation of the line $L$ .
18N.1.AHL.TZ0.H_5b:
Hence or otherwise, find the values of $t$ for which the angle between a and b is obtuse .
17M.2.AHL.TZ2.H_9e:
Verify that 2j + k is perpendicular to the plane Π $_2$ .
19M.1.SL.TZ2.S_2a:
parallel.
19N.1.SL.TZ0.S_9c:
Calculate the area of triangle ${\text{AOC}}$ .
17M.1.SL.TZ1.S_8c:
The lines ${L_1}$ and ${L_1}$ intersect at $C(9,{\text{ }}13,{\text{ }}z)$ . Find $z$ .
18M.1.SL.TZ1.S_9c.i:
Find $\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to$ .
17M.1.AHL.TZ1.H_5b:
By using a suitable scalar product of two vectors, determine whether ${\rm{A\hat BC}}$ is acute or obtuse.
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Show that $l_{1}$ and $l_{2}$ are never perpendicular to each other.
17M.1.SL.TZ1.S_8b:
A second line ${L_2}$ , has equation r = $\left( {\begin{array}{*{20}{c}} 1 \\ {13} \\ { - 14} \end{array}} \right) + s\left( {\begin{array}{*{20}{c}} p \\ 0 \\ 1 \end{array}} \right)$ .

Given that ${L_1}$ and ${L_2}$ are perpendicular, show that $p = 2$ .
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Find the volume of right-pyramid $ABCD$ .
21M.1.AHL.TZ2.8a:
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21M.1.AHL.TZ2.8b:
Find the minimum distance between $l_{1}$ and $l_{2}$ .
21N.2.AHL.TZ0.11b:
The line $L$ is the intersection of $Π_{1}$ and $Π_{2}$ . Verify that the vector equation of $L$ can be written as $r = (\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}) + λ (\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix})$ .

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Topic 3— Geometry and trigonometry » AHL 3.14—Vector equation of line

Topic 3— Geometry and trigonometry