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Date	May 2018	Marks available	6	Reference code	18M.1.SL.TZ1.S_6
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	S_6	Adapted from	N/A

Question

Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.
This is shown in the following diagram.

The vectors p , q and r are shown on the diagram.

Find p•(p + q + r).

Markscheme

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

METHOD 1 (using |p| |2q| cosθ)

finding p + q + r (A1)

eg 2q,

| p + q + r | = 2 × 3 (= 6) (seen anywhere) A1

correct angle between p and q (seen anywhere) (A1)

$\frac{\pi }{3}$ (accept 60°)

substitution of their values (M1)

eg 3 × 6 × cos $\left( {\frac{\pi }{3}} \right)$

correct value for cos $\left( {\frac{\pi }{3}} \right)$ (seen anywhere) (A1)

eg $\frac{1}{2},\,\,\,3 \times 6 \times \frac{1}{2}$

p•(p + q + r) = 9 A1 N3

METHOD 2 (scalar product using distributive law)

correct expression for scalar distribution (A1)

eg p• p + p•q + p•r

three correct angles between the vector pairs (seen anywhere) (A2)

eg 0° between p and p, $\frac{\pi }{3}$ between p and q, $\frac{{2\pi }}{3}$ between p and r

Note: Award A1 for only two correct angles.

substitution of their values (M1)

eg 3.3.cos0 +3.3.cos $\frac{\pi }{3}$ + 3.3.cos120

one correct value for cos0, cos $\left( {\frac{\pi }{3}} \right)$ or cos $\left( {\frac{2\pi }{3}} \right)$ (seen anywhere) A1

eg $\frac{1}{2},\,\,\,3 \times 6 \times \frac{1}{2}$

p•(p + q + r) = 9 A1 N3

METHOD 3 (scalar product using relative position vectors)

valid attempt to find one component of p or r (M1)

eg sin 60 = $\frac{x}{3}$ , cos 60 = $\frac{x}{3}$ , one correct value $\frac{3}{2},\,\,\frac{{3\sqrt 3 }}{2},\,\,\frac{{ - 3\sqrt 3 }}{2}$

one correct vector (two or three dimensions) (seen anywhere) A1

eg $p = \left( \begin{gathered} \,\,\,\frac{3}{2} \hfill \\ \frac{{3\sqrt 3 }}{2} \hfill \\ \end{gathered} \right),\,\,q = \left( \begin{gathered} 3 \hfill \\ 0 \hfill \\ \end{gathered} \right),\,\,r = \left( \begin{gathered} \,\,\,\,\frac{3}{2} \hfill \\ - \frac{{3\sqrt 3 }}{2} \hfill \\ \,\,\,\,0 \hfill \\ \end{gathered} \right)$

three correct vectors p + q + r = 2q (A1)

p + q + r = $\left( \begin{gathered} 6 \hfill \\ 0 \hfill \\ \end{gathered} \right)$ or $\left( \begin{gathered} 6 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)$ (seen anywhere, including scalar product) (A1)

correct working (A1)
eg $\left( {\frac{3}{2} \times 6} \right) + \left( {\frac{{3\sqrt 3 }}{2} \times 0} \right),\,\,9 + 0 + 0$

p•(p + q + r) = 9 A1 N3

[6 marks]

Examiners report

[N/A]

Syllabus sections

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

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22M.2.AHL.TZ2.11e:
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18M.1.SL.TZ1.S_9b.i:
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17M.1.SL.TZ1.S_8a.ii:
Hence, write down a vector equation for ${L_1}$ .
EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .
19M.2.SL.TZ2.S_7:
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19M.1.SL.TZ2.S_2b:
perpendicular.
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18N.2.SL.TZ0.S_8a.ii:
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19M.1.AHL.TZ1.H_1:
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Given that a and b are perpendicular, find the possible values of $k$ .
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18N.2.SL.TZ0.S_8b.i:
Find the value of $y$ .
18M.2.SL.TZ2.S_8a.i:
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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Find the vector $\vec{AB}$ and the vector $\vec{AC}$ .
21N.2.AHL.TZ0.11a.ii:
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Find the reflection of the point $B$ in the plane $Π_{3}$ .
21N.2.AHL.TZ0.11d.ii:
Hence find the vector equation of the line formed when $L$ is reflected in the plane $Π_{3}$ .
21N.2.AHL.TZ0.11c.i:
Show that at the point $P, λ = \frac{3}{4}$ .
21N.2.AHL.TZ0.11c.ii:
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Hence or otherwise find the shortest distance from R to the line through P and Q.
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Let w = u − v. Find the magnitude of w.
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Write down the value of angle OBA.
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Find the acute angle between the planes Π $_2$ and Π $_3$ .
18M.1.SL.TZ1.S_9b.ii:
Point C (k , 12 , −k) is on L. Show that k = 14.
17N.2.SL.TZ0.S_3b:
Let $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 0 \\ 0 \end{array}} \right)$ . Find ${\rm{B\hat AC}}$ .
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.
EXN.2.AHL.TZ0.11c:
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.
EXN.1.AHL.TZ0.8a:
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 $p = 2$ .
EXN.2.AHL.TZ0.11e:
Find the volume of right-pyramid $ABCD$ .
21M.1.AHL.TZ2.8a:
Show that $l_{1}$ and $l_{2}$ do not intersect.
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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