IB Questionbank

User interface language: English | Español

Date	May 2019	Marks available	1	Reference code	19M.1.AHL.TZ2.H_2
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Find	Question number	H_2	Adapted from	N/A

Question

Three points in three-dimensional space have coordinates A(0, 0, 2), B(0, 2, 0) and C(3, 1, 0).

Find the vector $- - \to AB$ $\overrightarrow {{\text{AB}}}$ .

[1]

a.i.

Find the vector $- - \to AC$ $\overrightarrow {{\text{AC}}}$ .

[1]

a.ii.

Hence or otherwise, find the area of the triangle ABC.

[4]

Markscheme

$- - \to AB = ⎛ ⎜ ⎝ \begin{matrix} 02 - 2 \end{matrix} ⎞ ⎟ ⎠$ $\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 0 \\ 2 \\ { - 2} \end{array}} \right)$ A1

Note: Accept row vectors or equivalent.

[1 mark]

a.i.

$- - \to AC = ⎛ ⎜ ⎝ \begin{matrix} 31 - 2 \end{matrix} ⎞ ⎟ ⎠$ $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ { - 2} \end{array}} \right)$ A1

Note: Accept row vectors or equivalent.

[1 mark]

a.ii.

METHOD 1

attempt at vector product using $- - \to AB$ $\overrightarrow {{\text{AB}}}$ and $- - \to AC$ $\overrightarrow {{\text{AC}}}$ . (M1)

±(2i + 6j +6k) A1

attempt to use area $= \frac{1}{2} ∣ ∣ ∣ - - \to AB \times - - \to AC ∣ ∣ ∣$ $= \frac{1}{2}\left| {\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AC}}} } \right|$ M1

$= \frac{\sqrt{76}}{2} (= \sqrt{19})$ $= \frac{{\sqrt {76} }}{2}\,\,\,\left( { = \sqrt {19} } \right)$ A1

METHOD 2

attempt to use $- - \to AB ∙ - - \to AC = ∣ ∣ ∣ - - \to AB ∣ ∣ ∣ ∣ ∣ ∣ - - \to AC ∣ ∣ ∣ cos θ$ $\overrightarrow {{\text{AB}}} \bullet \overrightarrow {{\text{AC}}} = \left| {\overrightarrow {{\text{AB}}} } \right|\left| {\overrightarrow {{\text{AC}}} } \right|{\text{cos}}\,\theta$ M1

$⎛ ⎜ ⎝ \begin{matrix} 02 - 2 \end{matrix} ⎞ ⎟ ⎠ \cdot ⎛ ⎜ ⎝ \begin{matrix} 31 - 2 \end{matrix} ⎞ ⎟ ⎠ = \sqrt{0^{2} + 2^{2} + {(- 2)}^{2}} \sqrt{3^{2} + 1^{2} + {(- 2)}^{2}} cos θ$ $\left( {\begin{array}{*{20}{c}} 0 \\ 2 \\ { - 2} \end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ { - 2} \end{array}} \right) = \sqrt {{0^2} + {2^2} + {{\left( { - 2} \right)}^2}} \sqrt {{3^2} + {1^2} + {{\left( { - 2} \right)}^2}} {\text{cos}}\,\theta$

$6 = \sqrt{8} \sqrt{14} cos θ$ $6 = \sqrt 8 \sqrt {14} \,{\text{cos}}\,\theta$ A1

$cos θ = \frac{6}{\sqrt{8} \sqrt{14}} = \frac{6}{\sqrt{112}}$ ${\text{cos}}\,\theta = \frac{6}{{\sqrt 8 \sqrt {14} }} = \frac{6}{{\sqrt {112} }}$

attempt to use area $= \frac{1}{2} ∣ ∣ ∣ - - \to AB \times - - \to AC ∣ ∣ ∣ sin θ$ $= \frac{1}{2}\left| {\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AC}}} } \right|{\text{sin}}\,\theta$ M1

$= \frac{1}{2} \sqrt{8} \sqrt{14} \sqrt{1 - \frac{36}{112}} (= \frac{1}{2} \sqrt{8} \sqrt{14} \sqrt{\frac{76}{112}})$ $= \frac{1}{2}\sqrt 8 \sqrt {14} \sqrt {1 - \frac{{36}}{{112}}} \,\left( { = \frac{1}{2}\sqrt 8 \sqrt {14} \sqrt {\frac{{76}}{{112}}} } \right)$

$= \frac{\sqrt{76}}{2} (= \sqrt{19})$ $= \frac{{\sqrt {76} }}{2}\,\,\,\left( { = \sqrt {19} } \right)$ A1

[4 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.13—Scalar and vector products

Show 37 related questions

18M.2.SL.TZ2.S_8b:
Find the angle between PQ and PR.
22M.1.AHL.TZ1.6c:
Find $FÂO$ $FÂO$ , the angle the rope makes with the platform.
22M.1.AHL.TZ2.16b:
Show that the acceleration vector of $P$ $P$ is never parallel to the position vector of $P$ $P$ .
22M.1.AHL.TZ2.16a:
Find the velocity vector of $P$ $P$ .
21N.1.AHL.TZ0.16a:
Find the position vector $- \to OS$ $\vec{OS}$ of the ship at time $t$ $t$ hours.
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).
18M.2.SL.TZ2.S_8a.i:
Find $\to PQ$ $\mathop {{\text{PQ}}}\limits^ \to$ .
19M.1.SL.TZ1.S_6:
The magnitudes of two vectors, u and v, are 4 and $\sqrt{3}$ $\sqrt 3$ respectively. The angle between u and v is $\frac{π}{6}$ $\frac{\pi }{6}$ .

Let w = u − v. Find the magnitude of w.
18M.2.SL.TZ2.S_8d:
Hence or otherwise find the shortest distance from R to the line through P and Q.
16N.1.SL.TZ0.S_4a:
Find a vector equation of the line that passes through P and Q.
16N.1.SL.TZ0.S_4b:
The line through P and Q is perpendicular to the vector 2i $+$ $+$ nk. Find the value of $n$ $n$ .
18N.1.AHL.TZ0.H_5b:
Hence or otherwise, find the values of $t$ $t$ for which the angle between a and b is obtuse .
SPM.1.AHL.TZ0.11a:
Given that v is perpendicular to B, find the value of $d$ $d$ .
19M.1.AHL.TZ1.H_1:
Let a = $⎛ ⎜ ⎝ \begin{matrix} 2 k - 1 \end{matrix} ⎞ ⎟ ⎠$ $\left( {\begin{array}{*{20}{c}} 2 \\ k \\ { - 1} \end{array}} \right)$ and b = $⎛ ⎜ ⎝ \begin{matrix} - 3 k + 2 k \end{matrix} ⎞ ⎟ ⎠$ $\left( {\begin{array}{*{20}{c}} { - 3} \\ {k + 2} \\ k \end{array}} \right)$ , $k \in \mathbb{R}$ .

Given that a and b are perpendicular, find the possible values of $k$ .
17N.2.SL.TZ0.S_3a:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
EXN.2.AHL.TZ0.7c.iii:
How many complete revolutions has the ball completed from $t = 0$ to the instant at which the string breaks?
19M.1.AHL.TZ2.H_2b:
Hence or otherwise, find the area of the triangle ABC.
SPM.1.AHL.TZ0.11b:
The force, F, produced by P moving in the magnetic field is given by the vector equation F = $a$ v × B, $a \in {\mathbb{R}^ + }$ .

Given that | F | = 14, find the value of $a$ .
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.
19M.1.AHL.TZ2.H_2a.ii:
Find the vector $\overrightarrow {{\text{AC}}}$ .
18M.2.SL.TZ2.S_8c:
Find the area of triangle PQR.
18M.2.SL.TZ2.S_8a.ii:
Find $\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|$ .
18N.1.AHL.TZ0.H_5a:
Find and simplify an expression for a • b in terms of $t$ .
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.
18M.1.SL.TZ2.S_1a:
Find a vector equation for L₁.
19M.1.SL.TZ2.S_2a:
parallel.
17N.2.SL.TZ0.S_3b:
Let $\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 0 \\ 0 \end{array}} \right)$ . Find ${\rm{B\hat AC}}$ .
EXN.2.AHL.TZ0.7b.i:
Find an expression for the velocity of the ball at time $t$ .
21M.1.AHL.TZ2.8a:
Find the possible value(s) for $p$ .
21M.1.AHL.TZ2.8b:
In the case that $p < 0$ , determine whether the lines intersect.
EXN.2.AHL.TZ0.7b.ii:
Hence show that the velocity of the ball is always perpendicular to the position vector of the ball.
EXN.2.AHL.TZ0.7c.i:
Find an expression for the acceleration of the ball at time $t$ .
EXN.2.AHL.TZ0.7a:
Show that the ball is moving in a circle with its centre at $O$ and state the radius of the circle.
EXN.2.AHL.TZ0.7c.ii:
Find the value of $t$ at the instant the string breaks.
21N.1.AHL.TZ0.16b:
Find the value of $t$ when the ship will be closest to the lighthouse.
21N.1.AHL.TZ0.16c:
An alarm will sound if the ship travels within $20$ kilometres of the lighthouse.

State whether the alarm will sound. Give a reason for your answer.

Hide related questions

Topic 3—Geometry and trigonometry

Question

Markscheme

Examiners report

Syllabus sections

View options