IB Questionbank

Updates to Questionbank

User interface language: English | Español

Date	May 2018	Marks available	2	Reference code	18M.2.SL.TZ2.S_8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	Question number	S_8	Adapted from	N/A

Question

Two points P and Q have coordinates (3, 2, 5) and (7, 4, 9) respectively.

Let ${\mathop {{\text{PR}}}\limits^ \to }$ = 6i − j + 3k.

Find $\mathop {{\text{PQ}}}\limits^ \to$ .

[2]

a.i.

Find $\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|$ .

[2]

a.ii.

Find the angle between PQ and PR.

[4]

b.

Find the area of triangle PQR.

[2]

c.

Hence or otherwise find the shortest distance from R to the line through P and Q.

[3]

d.

Markscheme

valid approach (M1)

eg (7, 4, 9) − (3, 2, 5) A − B

$\mathop {{\text{PQ}}}\limits^ \to =$ 4i + 2j + 4k $\left( { = \left( \begin{gathered} 4 \hfill \\ 2 \hfill \\ 4 \hfill \\ \end{gathered} \right)} \right)$ A1 N2

[2 marks]

a.i.

correct substitution into magnitude formula (A1)
eg $\sqrt {{4^2} + {2^2} + {4^2}}$

$\left| {\mathop {{\text{PQ}}}\limits^ \to } \right| = 6$ A1 N2

[2 marks]

a.ii.

finding scalar product and magnitudes (A1)(A1)

scalar product = (4 × 6) + (2 × (−1) + (4 × 3) (= 34)

magnitude of PR = $\sqrt {36 + 1 + 9} = \left( {6.782} \right)$

correct substitution of their values to find cos ${\text{Q}}\mathop {\text{P}}\limits^ \wedge {\text{R}}$ M1

eg cos ${\text{Q}}\mathop {\text{P}}\limits^ \wedge {\text{R}}\,\,{\text{ = }}\frac{{24 - 2 + 12}}{{\left( 6 \right) \times \left( {\sqrt {46} } \right)}},\,\,0.8355$

0.581746

${\text{Q}}\mathop {\text{P}}\limits^ \wedge {\text{R}}$ = 0.582 radians or ${\text{Q}}\mathop {\text{P}}\limits^ \wedge {\text{R}}$ = 33.3° A1 N3

[4 marks]

b.

correct substitution (A1)
eg $\frac{1}{2} \times \left| {\mathop {{\text{PQ}}}\limits^ \to } \right| \times \left| {\mathop {{\text{PR}}}\limits^ \to } \right| \times \,\,{\text{sin}}\,P,\,\,\frac{1}{2} \times 6 \times \sqrt {46} \times \,\,{\text{sin}}\,0.582$

area is 11.2 (sq. units) A1 N2

[2 marks]

c.

recognizing shortest distance is perpendicular distance from R to line through P and Q (M1)

eg sketch, height of triangle with base $\left[ {{\text{PQ}}} \right],\,\,\frac{1}{2} \times 6 \times h,\,\,{\text{sin}}\,33.3^\circ = \frac{h}{{\sqrt {46} }}$

correct working (A1)

eg $\frac{1}{2} \times 6 \times d = 11.2,\,\,\left| {\mathop {{\text{PR}}}\limits^ \to } \right| \times \,\,{\text{sin}}\,P,\,\,\sqrt {46} \times \,\,{\text{sin}}\,33.3^\circ$

3.72677

distance = 3.73 (units) A1 N2

[3 marks]

d.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.

[N/A]

c.

[N/A]

d.

Syllabus sections

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

Show 77 related questions

22M.2.AHL.TZ2.11c:
Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.
18M.1.SL.TZ1.S_9d:
Point D is also on L and has coordinates (8, 4, −9).

Find the area of triangle OCD.
17M.1.SL.TZ1.S_8d.ii:
Hence or otherwise, find one point on ${L_2}$ which is $\sqrt 5$ units from C.
19N.1.SL.TZ0.S_9a.i:
$\overrightarrow {{\text{OA}}} \bullet \overrightarrow {{\text{OC}}}$ .
18M.1.SL.TZ1.S_6:
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).
17M.2.AHL.TZ2.H_9f:
Find a vector perpendicular to the plane Π $_3$ .
17M.2.AHL.TZ2.H_9a:
Find the vector equation of the line (BC).
17M.1.AHL.TZ1.H_5a:
Find the area of the parallelogram ABCD.
18N.1.SL.TZ0.S_5:
Consider the vectors a = $\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right)$ and b = $\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)$ .

Find the possible values of p for which a and b are parallel.
17M.1.SL.TZ1.S_8d.i:
Find a unit vector in the direction of ${L_2}$ .
17M.2.AHL.TZ2.H_9d:
Show that the line (BC) lies in the plane Π $_1$ .
16N.1.SL.TZ0.S_4a:
Find a vector equation of the line that passes through P and Q.
18N.2.SL.TZ0.S_8d:
Find the area of triangle ABC.
17N.1.SL.TZ0.S_9a.i:
Show that $\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right)$ .
16N.1.SL.TZ0.S_4b:
The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$ .
18N.1.AHL.TZ0.H_5a:
Find and simplify an expression for a • b in terms of $t$ .
17N.1.SL.TZ0.S_9b:
Find the value of $p$ .
18M.1.SL.TZ2.S_1a:
Find a vector equation for L₁.
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:
Let $D (t)$ represent the distance between airplane $A$ and airplane $B$ for $0 \leq t \leq 2.5$ .

Find the minimum value of $D (t)$ .
18M.1.SL.TZ1.S_9b.i:
Find a vector equation for L.
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:
The vector equation of line $L$ is given by r $= \left( {\begin{array}{*{20}{c}} { - 1} \\ 3 \\ 8 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 4 \\ 5 \\ { - 1} \end{array}} \right)$ .

Point P is the point on $L$ that is closest to the origin. Find the coordinates of P.
19M.1.SL.TZ2.S_2b:
perpendicular.
18N.2.SL.TZ0.S_8a.i:
Find $\overrightarrow {{\text{AB}}}$ .
17M.2.AHL.TZ2.H_9c:
Find the Cartesian equation of the plane Π $_1$ , which passes through C and is perpendicular to $\overrightarrow {{\rm{OA}}}$ .
18N.2.SL.TZ0.S_8a.ii:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
18N.2.SL.TZ0.S_8c:
Find the angle between $\overrightarrow {{\text{AB}}}$ and $\overrightarrow {{\text{AC}}}$ .
18M.1.AHL.TZ2.H_1:
The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ.

Find cos θ.
17M.2.AHL.TZ2.H_9b:
Determine whether or not the lines (OA) and (BC) intersect.
19N.1.SL.TZ0.S_9b:
Given that ${\text{A}}\widehat {\text{O}}{\text{C}} = {\text{B}}\widehat {\text{O}}{\text{C}}$ , show that $k = 7$ .
18M.2.SL.TZ2.S_8a.ii:
Find $\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|$ .
EXN.2.AHL.TZ0.11a:
Find the vectors $\vec{AB}$ and $\vec{AC}$ .
17N.1.SL.TZ0.S_9a.ii:
Find a vector equation for $L$ .
21M.1.AHL.TZ2.5:
Given any two non-zero vectors, $a$ and $b$ , show that ${|a \times b|}^{2} = {|a|}^{2} {|b|}^{2} - {(a \cdot b)}^{2}$ .
18M.2.SL.TZ2.S_8b:
Find the angle between PQ and PR.
17N.2.SL.TZ0.S_3a:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
EXN.2.AHL.TZ0.11b:
Use a vector method to show that $B \hat{A} C = 60 °$ .
17M.1.SL.TZ1.S_8a.i:
Find $\overrightarrow {AB}$ .
19M.1.AHL.TZ1.H_1:
Let a = $\left( {\begin{array}{*{20}{c}} 2 \\ k \\ { - 1} \end{array}} \right)$ and b = $\left( {\begin{array}{*{20}{c}} { - 3} \\ {k + 2} \\ k \end{array}} \right)$ , $k \in \mathbb{R}$ .

Given that a and b are perpendicular, find the possible values of $k$ .
18M.1.SL.TZ1.S_9a:
Show that $\mathop {{\text{AB}}}\limits^ \to = \left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)$
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 $q$ .
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.2.SL.TZ2.S_8d:
Hence or otherwise find the shortest distance from R to the line through P and Q.
17M.1.SL.TZ2.S_2b:
Given that c = a + 2b, find c.
17M.1.SL.TZ2.S_2a:
Find the value of $k$ .
19M.1.SL.TZ1.S_6:
The magnitudes of two vectors, u and v, are 4 and $\sqrt 3$ respectively. The angle between u and v is $\frac{\pi }{6}$ .

Let w = u − v. Find the magnitude of w.
18M.1.SL.TZ1.S_9c.ii:
Write down the value of angle OBA.
17M.2.AHL.TZ2.H_9g:
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 $r = (\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}) + λ (\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix})$ .

Hide related questions

Topic 3— Geometry and trigonometry