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Date	November 2016	Marks available	5	Reference code	16N.2.AHL.TZ0.H_2
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	H_2	Adapted from	N/A

Question

Find the acute angle between the planes with equations $x + y + z = 3$ $x + y + z = 3$ and $2 x - z = 2$ $2x - z = 2$ .

Markscheme

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

n $_{1} = ⎛ ⎜ ⎝ \begin{matrix} 111 \end{matrix} ⎞ ⎟ ⎠$ $_1 = \left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right)$ and n $_{2} = ⎛ ⎜ ⎝ \begin{matrix} 20 - 1 \end{matrix} ⎞ ⎟ ⎠$ $_2 = \left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ { - 1} \end{array}} \right)$ (A1)(A1)

EITHER

$θ = arccos (\frac{n_{1} ∙ n_{2}}{| n_{1} | | n_{2} |}) (cos θ = \frac{n_{1} ∙ n_{2}}{| n_{1} | | n_{2} |})$ $\theta = \arccos \left( {\frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\cos \theta = \frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)$ (M1)

$= arccos (\frac{2 + 0 - 1}{\sqrt{3} \sqrt{5}}) (cos θ = \frac{2 + 0 - 1}{\sqrt{3} \sqrt{5}})$ $= \arccos \left( {\frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\cos \theta = \frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)$ (A1)

$= arccos (\frac{1}{\sqrt{15}}) (cos θ = \frac{1}{\sqrt{15}})$ $= \arccos \left( {\frac{1}{{\sqrt {15} }}} \right)\left( {\cos \theta = \frac{1}{{\sqrt {15} }}} \right)$

$θ = arcsin (\frac{| n_{1} \times n_{2} |}{| n_{1} | | n_{2} |}) (sin θ = \frac{| n_{1} \times n_{2} |}{| n_{1} | | n_{2} |})$ $\theta = \arcsin \left( {\frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\sin \theta = \frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)$ (M1)

$= arcsin (\frac{\sqrt{14}}{\sqrt{3} \sqrt{5}}) (sin θ = \frac{\sqrt{14}}{\sqrt{3} \sqrt{5}})$ $= \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\sin \theta = \frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)$ (A1)

$= arcsin (\frac{\sqrt{14}}{\sqrt{15}}) (sin θ = \frac{\sqrt{14}}{\sqrt{15}})$ $= \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)\left( {\sin \theta = \frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)$

THEN

$= {75.0}^{\circ} (or 1.31)$ $= 75.0^\circ {\text{ (or 1.31)}}$ A1

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.18—Intersections of lines & planes

Show 55 related questions

22M.1.AHL.TZ1.11b.i:
Verify that the point $P (1, - 2, 0)$ $P (1, - 2, 0)$ lies on both $\prod_{1}$ $\prod_{1}$ and $\prod_{2}$ $\prod_{2}$ .
16N.1.AHL.TZ0.H_1:
Find the coordinates of the point of intersection of the planes defined by the equations $x + y + z = 3, x - y + z = 5$ $x + y + z = 3,{\text{ }}x - y + z = 5$ and $x + y + 2 z = 6$ $x + y + 2z = 6$ .
18M.1.AHL.TZ2.H_9a.i:
Explain why ABCD is a parallelogram.
18M.1.AHL.TZ2.H_9e:
Find the Cartesian equation of $Π$ $\Pi$ .
18M.1.AHL.TZ1.H_10c:
Find the Cartesian equation of $Π_{3}$ ${\Pi _3}$ .
19N.1.AHL.TZ0.H_11b.ii:
Comment on the positions of $V$ ${\text{V}}$ in relation to the plane $ABC$ ${\text{ABC}}$ .
19N.1.AHL.TZ0.H_11c.ii:
Find the equation of the horizontal asymptote of the graph.
19N.1.AHL.TZ0.H_8:
A straight line, $L_{θ}$ ${L_\theta }$ , has vector equation r $= \left( {\begin{array}{*{20}{c}} 5 \\ 0 \\ 0 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 5 \\ {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right){\text{, }}\lambda {\text{, }}\theta \in \mathbb{R}$ .

The plane ${\Pi _p}$ , has equation $x = p{\text{, }}p \in \mathbb{R}$ .

Show that the angle between ${L_\theta }$ and ${\Pi _p}$ is independent of both $\theta$ and $p$ .
18M.1.AHL.TZ2.H_9d:
Find the vector equation of the straight line passing through M and normal to the plane $\Pi$ containing ABCD.
18M.1.AHL.TZ1.H_10b:
Find the angle between the faces ABD and BCD.
19M.2.AHL.TZ2.H_11d.ii:
Given that П₃ is equally inclined to both П₁ and П₂, determine two distinct possible Cartesian equations for П₃.
16N.1.AHL.TZ0.H_8a:
find the value of $a$ ;
22M.1.AHL.TZ1.11b.ii:
Find a vector equation of $L$ , the line of intersection of $\prod_{1}$ and $\prod_{2}$ .
22M.1.AHL.TZ1.11c:
Find the distance between $L$ and $\prod_{3}$ .
21M.2.AHL.TZ1.6a:
Find a Cartesian equation of the plane $Π_{3}$ which is perpendicular to $Π_{1}$ and $Π_{2}$ and passes through the origin $(0, 0, 0)$ .
EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .
19N.1.AHL.TZ0.H_3:
Three planes have equations:

$2x - y + z = 5$

$x + 3y - z = 4$ , where $a{\text{, }}b \in \mathbb{R}$ .

$3x - 5y + az = b$

Find the set of values of $a$ and $b$ such that the three planes have no points of intersection.
18N.1.AHL.TZ0.H_9b:
Find, in terms of $b$ , the equation of the line L which passes through M and is perpendicular to the plane П.
19M.2.AHL.TZ2.H_11a:
Find the Cartesian equation of the plane containing P, Q and R.
18M.1.AHL.TZ2.H_9a.ii:
Using vector algebra, show that $\mathop {{\text{AD}}}\limits^ \to = \mathop {{\text{BC}}}\limits^ \to$ .
20N.2.AHL.TZ0.H_10c:
Find the equation of $Π_{2}$ , giving your answer in the form $r . n = d$ .
19M.2.AHL.TZ2.H_11d.i:
Show that $5a - 7c = 4$ .
19N.1.AHL.TZ0.H_11b.i:
Find the two possible coordinates of ${\text{V}}$ .
18M.1.AHL.TZ1.H_10a:
Find the Cartesian equation of the plane ${\Pi _1}$ , passing through the points A , B and D.
16N.1.AHL.TZ0.H_8b:
determine the coordinates of the point of intersection P.
20N.2.AHL.TZ0.H_10b:
Find the shortest distance from the point $O (0, 0, 0)$ to $Π_{1}$ .
18M.1.AHL.TZ2.H_9b:
Show that p = 1, q = 1 and r = 4.
EXN.2.AHL.TZ0.11a:
Find the vectors $\vec{AB}$ and $\vec{AC}$ .
18M.1.AHL.TZ1.H_10d:
Show that P is the midpoint of AD.
EXN.2.AHL.TZ0.11b:
Use a vector method to show that $B \hat{A} C = 60 °$ .
19M.2.AHL.TZ2.H_11b:
Given that П₁ and П₂ meet in a line $L$ , verify that the vector equation of $L$ can be given by r $= \left( {\begin{array}{*{20}{c}} {\frac{5}{4}} \\ 0 \\ { - \frac{7}{4}} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} {\frac{1}{2}} \\ 1 \\ { - \frac{5}{2}} \end{array}} \right)$ .
20N.2.AHL.TZ0.H_10a:
Given that $L$ meets $Π_{1}$ at the point $P$ , find the coordinates of $P$ .
17N.1.AHL.TZ0.H_2b:
Find the coordinates of the point of intersection of the line L with the plane Π.
22M.1.AHL.TZ1.11a:
Show that the three planes do not intersect.
21N.2.AHL.TZ0.11a.i:
Find the vector $\vec{AB}$ and the vector $\vec{AC}$ .
21N.2.AHL.TZ0.11a.ii:
Hence find the equation of $Π_{1}$ , expressing your answer in the form $a x + b y + c z = d$ , where $a, b, c, d \in ℤ$ .
21N.2.AHL.TZ0.11d.i:
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:
Hence find the coordinates of $P$ .
18M.1.AHL.TZ2.H_9f.ii:
Find YZ.
18M.1.AHL.TZ2.H_9f.i:
Find the coordinates of X, Y and Z.
18M.1.AHL.TZ2.H_9c:
Find the area of the parallelogram ABCD.
18M.1.AHL.TZ1.H_10e:
Find the area of the triangle OPQ.
20N.2.AHL.TZ0.H_10d:
Determine the acute angle between $Π_{1}$ and $Π_{2}$ .
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 $L$ .
19N.1.AHL.TZ0.H_11c.i:
At ${\text{X}}$ , find the value of $p$ and the value of $\theta$ .
17N.1.AHL.TZ0.H_2a:
Find a vector equation of the line L passing through the points A and B.
21M.2.AHL.TZ1.6b:
Find the coordinates of the point where $Π_{1}$ , $Π_{2}$ and $Π_{3}$ intersect.
18N.1.AHL.TZ0.H_9c:
Show that L does not intersect the $y$ -axis for any negative value of $b$ .
19M.2.AHL.TZ2.H_11c:
Given that П₃ is parallel to the line $L$ , show that $a + 2b - 5c = 0$ .
18N.1.AHL.TZ0.H_9a:
Find, in terms of $b$ , a Cartesian equation of the plane Π containing this triangle.
EXN.2.AHL.TZ0.11e:
Find the volume of right-pyramid $ABCD$ .
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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