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Date	May 2017	Marks available	2	Reference code	17M.2.AHL.TZ2.H_9
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Show that	Question number	H_9	Adapted from	N/A

Question

The points A, B and C have the following position vectors with respect to an origin O.

$\overrightarrow {{\rm{OA}}} = 2$ i + j – 2k

$\overrightarrow {{\rm{OB}}} = 2$ i – j + 2k

$\overrightarrow {{\rm{OC}}} =$ i + 3j + 3k

The plane Π $_2$ contains the points O, A and B and the plane Π $_3$ contains the points O, A and C.

Find the vector equation of the line (BC).

[3]

Determine whether or not the lines (OA) and (BC) intersect.

[6]

Find the Cartesian equation of the plane Π $_1$ , which passes through C and is perpendicular to $\overrightarrow {{\rm{OA}}}$ .

[3]

Show that the line (BC) lies in the plane Π $_1$ .

[2]

Verify that 2j + k is perpendicular to the plane Π $_2$ .

[3]

Find a vector perpendicular to the plane Π $_3$ .

[1]

Find the acute angle between the planes Π $_2$ and Π $_3$ .

[4]

Markscheme

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

$\overrightarrow {{\rm{BC}}}$ = (i + 3j + 3k) $-$ (2i $-$ j + 2k) = $-$ i + 4j + k (A1)

r = (2i $-$ j + 2k) + $\lambda$ ( $-$ i + 4j + k)

(or r = (i + 3j + 3k) + $\lambda$ ( $-$ i + 4j + k) (M1)A1

Note: Do not award A1 unless r = or equivalent correct notation seen.

[3 marks]

attempt to write in parametric form using two different parameters AND equate M1

$2\mu = 2 - \lambda$

$\mu = - 1 + 4\lambda$

$- 2\mu = 2 + \lambda$ A1

attempt to solve first pair of simultaneous equations for two parameters M1

solving first two equations gives $\lambda = \frac{4}{9},{\text{ }}\mu = \frac{7}{9}$ (A1)

substitution of these two values in third equation (M1)

since the values do not fit, the lines do not intersect R1

Note: Candidates may note that adding the first and third equations immediately leads to a contradiction and hence they can immediately deduce that the lines do not intersect.

[6 marks]

METHOD 1

plane is of the form r $\bullet$ (2i + j $-$ 2k) = d (A1)

d = (i + 3j + 3k) $\bullet$ (2i + j $-$ 2k) = $-$ 1 (M1)

hence Cartesian form of plane is $2x + y - 2z = - 1$ A1

METHOD 2

plane is of the form $2x + y - 2z = d$ (A1)

substituting $(1,{\text{ }}3,{\text{ }}3)$ (to find gives $2 + 3 - 6 = - 1$ ) (M1)

hence Cartesian form of plane is $2x + y - 2z = - 1$ A1

[3 marks]

METHOD 1

attempt scalar product of direction vector BC with normal to plane M1

( $-$ i + 4j + k) $\bullet$ (2i + j $-$ 2k) $= - 2 + 4 - 2$

$= 0$ A1

hence BC lies in Π $_1$ AG

METHOD 2

substitute eqn of line into plane M1

${\text{line }}r = \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 2 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} { - 1} \\ 4 \\ 1 \end{array}} \right).{\text{ Plane }}{\pi _1}:2x + y - 2z = - 1$

$2(2 - \lambda ) + ( - 1 + 4\lambda ) - 2(2 + \lambda )$

$= - 1$ A1

hence BC lies in Π $_1$ AG

Note: Candidates may also just substitute $2i - j + 2k$ into the plane since they are told C lies on ${\pi _1}$ .

Note: Do not award A1FT.

[2 marks]

METHOD 1

applying scalar product to $\overrightarrow {{\rm{OA}}}$ and $\overrightarrow {{\rm{OB}}}$ M1

(2j + k) $\bullet$ (2i + j $-$ 2k) = 0 A1

(2j + k) $\bullet$ (2i $-$ j + 2k) =0 A1

METHOD 2

attempt to find cross product of $\overrightarrow {{\rm{OA}}}$ and $\overrightarrow {{\rm{OB}}}$ M1

plane Π $_2$ has normal $\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}}$ = $-$ 8j $-$ 4k A1

since $-$ 8j $-$ 4k = $-$ 4(2j + k), 2j + k is perpendicular to the plane Π $_2$ R1

[3 marks]

plane Π $_3$ has normal $\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OC}}}$ = 9i $-$ 8j + 5k A1

[1 mark]

attempt to use dot product of normal vectors (M1)

$\cos \theta = \frac{{(2j + k) \bullet (9i - 8j + 5k)}}{{\left| {2j + k} \right|\left| {9i - 8j + 5k} \right|}}$ (M1)

$= \frac{{ - 11}}{{\sqrt 5 \sqrt {170} }}\,\,\,( = - 0.377 \ldots )$ (A1)

Note: Accept $\frac{{11}}{{\sqrt 5 \sqrt {170} }}$ . acute angle between planes $= 67.8^\circ \,\,\,{\text{(}} = 1.18^\circ )$ A1

[4 marks]

Examiners report

[N/A]

Syllabus sections

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

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16N.1.SL.TZ0.S_4a:
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18M.1.SL.TZ2.S_1a:
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19N.1.SL.TZ0.S_9a.ii:
$\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}}$ .
22M.2.AHL.TZ1.7c:
Find $q$ such that $| q | = | b |$ and $q$ is perpendicular to $a$ .
18N.2.SL.TZ0.S_8b.ii:
Show that $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 8 \\ { - 10} \\ { - 1} \end{array}} \right)$ .
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.1.SL.TZ2.S_2b:
perpendicular.
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Find the Cartesian equation of the plane Π $_1$ , which passes through C and is perpendicular to $\overrightarrow {{\rm{OA}}}$ .
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18M.1.AHL.TZ2.H_1:
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18M.2.SL.TZ2.S_8a.ii:
Find $\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|$ .
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Find the vectors $\vec{AB}$ and $\vec{AC}$ .
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17M.1.SL.TZ1.S_8a.i:
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19M.1.AHL.TZ1.H_1:
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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 reflection of the point $B$ in the plane $Π_{3}$ .
21N.2.AHL.TZ0.11d.ii:
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21N.2.AHL.TZ0.11c.i:
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21N.2.AHL.TZ0.11c.ii:
Hence find the coordinates of $P$ .
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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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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.
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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:
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17M.2.AHL.TZ2.H_9e:
Verify that 2j + k is perpendicular to the plane Π $_2$ .
19M.1.SL.TZ2.S_2a:
parallel.
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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 $z$ .
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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 $ABCD$ .
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Find the minimum distance between $l_{1}$ and $l_{2}$ .
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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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Markscheme

Examiners report

Syllabus sections

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