IB Questionbank

User interface language: English | Español

Date	May 2022	Marks available	2	Reference code	22M.2.AHL.TZ1.7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	7	Adapted from	N/A

Question

Consider the vectors $a$ and $b$ such that $a = (\begin{matrix} 12 \\ - 5 \end{matrix})$ and $|b| = 15$ .

Consider the vector $p$ such that $p = a + b$ .

Consider the vector $q$ such that $q = (\begin{matrix} x \\ y \end{matrix})$ , where $x, y \in ℝ^{+}$ .

Find the possible range of values for $|a + b|$ .

[2]

Given that $|a + b|$ is a minimum, find $p$ .

[2]

Find $q$ such that $| q | = | b |$ and $q$ is perpendicular to $a$ .

[5]

Markscheme

$|a| = \sqrt{12^{2} + {(- 5)}^{2}} (= 13)$ (A1)

$2 \leq |a + b| \leq 28$ (accept min $2$ and max $28$ ) A1

Note: Award (A1)A0 for $2$ and $28$ seen with no indication that they are the endpoints of an interval.

[2 marks]

recognition that $p$ or $b$ is a negative multiple of $a$ (M1)

$p = - 2 \hat{a}$ OR $b = - \frac{15}{13} a (= - \frac{15}{13} (\begin{matrix} 12 \\ - 5 \end{matrix}))$

$p = - \frac{2}{13} (\begin{matrix} 12 \\ - 5 \end{matrix}) (= (\begin{matrix} - 1.85 \\ 0.769 \end{matrix}))$ A1

[2 marks]

METHOD 1

$q$ is perpendicular to $(\begin{matrix} 12 \\ - 5 \end{matrix})$

$\Rightarrow q$ is in the direction $(\begin{matrix} 5 \\ 12 \end{matrix})$ (M1)

$q = k (\begin{matrix} 5 \\ 12 \end{matrix})$ (A1)

$(|q| =) \sqrt{{(5 k)}^{2} + {(12 k)}^{2}} = 15$ (M1)

$k = \frac{15}{13}$ (A1)

$q = \frac{15}{13} (\begin{matrix} 5 \\ 12 \end{matrix}) (= (\begin{matrix} \frac{75}{13} \\ \frac{180}{13} \end{matrix}) = (\begin{matrix} 5.77 \\ 13.8 \end{matrix}))$ A1

METHOD 2

$q$ is perpendicular to $(\begin{matrix} 12 \\ - 5 \end{matrix})$

attempt to set scalar product $q . a = 0$ OR product of gradients $= - 1$ (M1)

$12 x - 5 y = 0$ (A1)

$(|q| =) \sqrt{x^{2} + y^{2}} = 15$

attempt to solve simultaneously to find a quadratic in $x$ or $y$ (M1)

$x^{2} + {(\frac{12 x}{5})}^{2} = 15^{2}$ OR ${(\frac{5 y}{12})}^{2} + y^{2} = 15^{2}$

$q = (\begin{matrix} \frac{75}{13} \\ \frac{180}{13} \end{matrix}) (= (\begin{matrix} 5.77 \\ 13.8 \end{matrix}))$ A1A1

Note: Award A1 independently for each value. Accept values given as $x = \frac{75}{13}$ and $y = \frac{180}{13}$ or equivalent.

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.12—Vector definitions

Show 77 related questions

18M.2.AHL.TZ1.H_11c.iii:
Find the distance between the two submarines at this time.
18M.2.AHL.TZ1.H_11b.ii:
Find the value of t when submarine B passes through P.
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.
17N.1.AHL.TZ0.H_9a.i:
Find, in terms of a and b $\overrightarrow {{\text{OF}}}$ .
18M.1.AHL.TZ2.H_9a.i:
Explain why ABCD is a parallelogram.
18M.1.AHL.TZ2.H_9e:
Find the Cartesian equation of $\Pi$ .
19M.1.AHL.TZ1.H_11d:
Write down $\overrightarrow {{\text{PA}}}$ and $\overrightarrow {{\text{PB}}}$ .
18M.2.AHL.TZ1.H_11c.ii:
Find the value of t when the two submarines are closest together.
19N.1.SL.TZ0.S_9a.i:
$\overrightarrow {{\text{OA}}} \bullet \overrightarrow {{\text{OC}}}$ .
19M.1.AHL.TZ1.H_11a.i:
Find how many sets of four points can be selected which can form the vertices of a quadrilateral.
19N.2.SL.TZ0.S_2b:
Write down a direction vector for ${L_3}$ .
19M.1.AHL.TZ1.H_11c:
Write down the value of $\lambda$ corresponding to the point ${\text{A}}$ .
19M.1.AHL.TZ1.H_11a.ii:
Find how many sets of three points can be selected which can form the vertices of a triangle.
18M.2.AHL.TZ1.H_11c.i:
Find an expression for the distance between the two submarines in terms of t.
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.
17M.1.SL.TZ1.S_8d.i:
Find a unit vector in the direction of ${L_2}$ .
18M.2.AHL.TZ1.H_11a:
Show that the two submarines would collide at a point P and write down the coordinates of P.
18N.2.SL.TZ0.S_8d:
Find the area of triangle ABC.
19M.1.AHL.TZ2.H_2a.i:
Find the vector $\overrightarrow {{\text{AB}}}$ .
19M.1.AHL.TZ2.H_2a.ii:
Find the vector $\overrightarrow {{\text{AC}}}$ .
19N.1.SL.TZ0.S_9a.ii:
$\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}}$ .
22M.2.AHL.TZ1.7a:
Find the possible range of values for $|a + b|$ .
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.TZ2.11b:
Show that airplane $A$ travels at a greater speed than airplane $B$ .
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 ${L_1}$ .
17N.1.AHL.TZ0.H_9b.i:
Find an expression for $\overrightarrow {{\text{OD}}}$ in terms of a, b and $\lambda$ ;
EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .
19M.1.SL.TZ2.S_2b:
perpendicular.
18N.2.SL.TZ0.S_8a.i:
Find $\overrightarrow {{\text{AB}}}$ .
18M.1.AHL.TZ2.H_9a.ii:
Using vector algebra, show that $\mathop {{\text{AD}}}\limits^ \to = \mathop {{\text{BC}}}\limits^ \to$ .
20N.1.SL.TZ0.S_9b:
Find the value of $m$ .
18N.2.SL.TZ0.S_8a.ii:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
19N.2.SL.TZ0.S_2c:
${L_3}$ passes through the intersection of ${L_1}$ and ${L_2}$ .

Write down a vector equation for ${L_3}$ .
18N.2.SL.TZ0.S_8c:
Find the angle between $\overrightarrow {{\text{AB}}}$ and $\overrightarrow {{\text{AC}}}$ .
17N.1.AHL.TZ0.H_9b.ii:
Find an expression for $\overrightarrow {{\text{OD}}}$ in terms of a, b and $\mu$ .
19M.1.AHL.TZ2.H_2b:
Hence or otherwise, find the area of the triangle ABC.
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.1.AHL.TZ2.H_9b:
Show that p = 1, q = 1 and r = 4.
19M.1.AHL.TZ1.H_11e:
Let ${\text{C}}$ be the point on ${l_1}$ with coordinates (1, 0, 1) and ${\text{D}}$ be the point on ${l_2}$ with parameter $\mu = - 2$ .

Find the area of the quadrilateral ${\text{CDBA}}$ .
EXN.2.AHL.TZ0.11a:
Find the vectors $\vec{AB}$ and $\vec{AC}$ .
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 $\overrightarrow {AB}$ .
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 $y$ .
19N.2.SL.TZ0.S_2a:
Find the point of intersection of ${L_1}$ and ${L_2}$ .
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.
19M.1.AHL.TZ1.H_11b:
Verify that ${\text{P}}$ is the point of intersection of the two lines.
18M.1.SL.TZ1.S_9c.ii:
Write down the value of angle OBA.
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.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.
17N.1.AHL.TZ0.H_9d:
Deduce an expression for $\overrightarrow {{\text{CD}}}$ in terms of a and b only.
20N.1.SL.TZ0.S_9a:
Express $\vec{AB}$ in terms of $m$ .
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 $L$ .
17N.1.AHL.TZ0.H_9e:
Given that area $\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})$ , find the value of $k$ .
19M.1.SL.TZ2.S_2a:
parallel.
17N.1.AHL.TZ0.H_9c:
Show that $\mu = \frac{1}{{13}}$ , and find the value of $\lambda$ .
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.2.AHL.TZ1.H_11b.i:
Show that submarine B travels in the same direction as originally planned.
17N.1.AHL.TZ0.H_9a.ii:
Find, in terms of a and b $\overrightarrow {{\text{AF}}}$ .
18M.1.SL.TZ1.S_9c.i:
Find $\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to$ .
20N.1.SL.TZ0.S_9c:
Consider a unit vector $u$ , such that $u = p i - \frac{2}{3} j + \frac{1}{3} k$ , where $p > 0$ .

Point $C$ is such that $\vec{BC} = 9 u$ .

Find the coordinates of $C$ .
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 $ABCD$ .
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

Question

Markscheme

Examiners report

Syllabus sections

View options