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Date	May 2021	Marks available	7	Reference code	21M.1.AHL.TZ1.11
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	11	Adapted from	N/A

Question

Consider the line $L_{1}$ defined by the Cartesian equation $\frac{x + 1}{2} = y = 3 - z$ .

Consider a second line $L_{2}$ defined by the vector equation $r = (\begin{matrix} 0 \\ 1 \\ 2 \end{matrix}) + t (\begin{matrix} a \\ 1 \\ - 1 \end{matrix})$ , where $t \in ℝ$ and $a \in ℝ$ .

Show that the point $(- 1, 0, 3)$ lies on $L_{1}$ .

[1]

a.i.

Find a vector equation of $L_{1}$ .

[3]

a.ii.

Find the possible values of $a$ when the acute angle between $L_{1}$ and $L_{2}$ is $45 °$ .

[8]

b.

It is given that the lines $L_{1}$ and $L_{2}$ have a unique point of intersection, $A$ , when $a \neq k$ .

Find the value of $k$ , and find the coordinates of the point $A$ in terms of $a$ .

[7]

c.

Markscheme

$\frac{- 1 + 1}{2} = 0 = 3 - 3$ A1

the point $(- 1, 0, 3)$ lies on $L_{1}$ . AG

[1 mark]

a.i.

attempt to set equal to a parameter or rearrange cartesian form (M1)

$\frac{x + 1}{2} = y = 3 - z = λ \Rightarrow x = 2 λ - 1, y = λ, z = 3 - λ$ OR $\frac{x + 1}{2} = \frac{y - 0}{1} = \frac{z - 3}{- 1}$

correct direction vector $(\begin{matrix} 2 \\ 1 \\ - 1 \end{matrix})$ or equivalent seen in vector form (A1)

$r = (\begin{matrix} - 1 \\ 0 \\ 3 \end{matrix}) + λ (\begin{matrix} 2 \\ 1 \\ - 1 \end{matrix})$ (or equivalent) A1

Note: Award A0 if $= r$ is omitted.

[3 marks]

a.ii.

attempt to use the scalar product formula (M1)

$(\begin{matrix} 2 \\ 1 \\ - 1 \end{matrix}) ∙ (\begin{matrix} a \\ 1 \\ - 1 \end{matrix}) = (\pm) \sqrt{6} \sqrt{a^{2} + 2} \cos 45 °$ (A1)(A1)

Note: Award A1 for LHS and A1 for RHS

$2 a + 2 = \frac{(\pm) \sqrt{6} \sqrt{a^{2} + 2} \sqrt{2}}{2} (\Rightarrow 2 a + 2 = (\pm) \sqrt{3} \sqrt{a^{2} + 2})$ A1A1

Note: Award A1 for LHS and A1 for RHS

$4 a^{2} + 8 a + 4 = 3 a^{2} + 6$ A1

$a^{2} + 8 a - 2 = 0$ M1

attempt to solve their quadratic

$a = \frac{- 8 \pm \sqrt{64 + 8}}{2} = \frac{- 8 \pm \sqrt{72}}{2} (= - 4 \pm 3 \sqrt{2})$ A1

[8 marks]

b.

METHOD 1

attempt to equate the parametric forms of $L_{1}$ and $L_{2}$ (M1)

$\{\begin{matrix} 2 λ - 1 = t a \\ λ = 1 + t \\ 3 - λ = 2 - t \end{matrix}$ A1

attempt to solve equations by eliminating $λ$ or $t$ (M1)

$2 + 2 t - 1 = t a \Rightarrow 1 = t (a - 2)$ or $2 λ - 1 = (λ - 1) a \Rightarrow a - 1 = λ (a - 2)$

Solutions exist unless $a - 2 = 0$

$k = 2$ A1

Note: This A1 is independent of the following marks.

$t = \frac{1}{a - 2}$ or $λ = \frac{a - 1}{a - 2}$ A1

$A$ has coordinates $(\frac{a}{a - 2}, 1 + \frac{1}{a - 2}, 2 - \frac{1}{a - 2}) (= (\frac{a}{a - 2}, \frac{a - 1}{a - 2}, \frac{2 a - 5}{a - 2}))$ A2

Note: Award A1 for any two correct coordinates seen or final answer in vector form.

METHOD 2

no unique point of intersection implies direction vectors of $L_{1}$ and $L_{2}$ parallel

$k = 2$ A1

Note: This A1 is independent of the following marks.

attempt to equate the parametric forms of $L_{1}$ and $L_{2}$ (M1)

$\{\begin{matrix} 2 λ - 1 = t a \\ λ = 1 + t \\ 3 - λ = 2 - t \end{matrix}$ A1

attempt to solve equations by eliminating $λ$ or $t$ (M1)

$2 + 2 t - 1 = t a \Rightarrow 1 = t (a - 2)$ or $2 λ - 1 = (λ - 1) a \Rightarrow a - 1 = λ (a - 2)$

$t = \frac{1}{a - 2}$ or $λ = \frac{a - 1}{a - 2}$ A1

$A$ has coordinates $(\frac{a}{a - 2}, 1 + \frac{1}{a - 2}, 2 - \frac{1}{a - 2}) (= (\frac{a}{a - 2}, \frac{a - 1}{a - 2}, \frac{2 a - 5}{a - 2}))$ A2

Note: Award A1 for any two correct coordinates seen or final answer in vector form.

[7 marks]

c.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.

[N/A]

c.

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.14—Vector equation of line

Show 98 related questions

22M.1.AHL.TZ1.11b.i:
Verify that the point $P (1, - 2, 0)$ lies on both $\prod_{1}$ and $\prod_{2}$ .
17M.1.SL.TZ2.S_9d:
Hence or otherwise, find the distance between ${P_1}$ and ${P_2}$ two seconds after they leave A.
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.
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 $\overrightarrow {{\text{PB}}}$ .
17M.2.AHL.TZ2.H_9f:
Find a vector perpendicular to the plane Π $_3$ .
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.
17M.2.AHL.TZ2.H_9a:
Find the vector equation of the line (BC).
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 ${\text{A}}$ .
19N.1.AHL.TZ0.H_8:
A straight line, ${L_\theta }$ , 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 $p$ .
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.
17M.1.SL.TZ2.S_9a:
Find the coordinates of A.
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 ${L_2}$ .
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 $\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right)$ .
17M.1.SL.TZ2.S_9c:
Find $\cos {\rm{B\hat AC}}$ .
16N.1.SL.TZ0.S_4b:
The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$ .
17N.1.SL.TZ0.S_9b:
Find the value of $p$ .
18M.1.SL.TZ2.S_1a:
Find a vector equation for L₁.
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 $\prod_{3}$ .
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 ${L_1}$ .
EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .
19M.1.SL.TZ1.S_2b:
A second line, ${L_2}$ , is parallel to ${L_1}$ and passes through (1, 2, 3).

Write down a vector equation for ${L_2}$ .
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.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.
21M.1.AHL.TZ1.11a.i:
Show that the point $(- 1, 0, 3)$ lies on $L_{1}$ .
21M.1.AHL.TZ1.11b:
Find the possible values of $a$ when the acute angle between $L_{1}$ and $L_{2}$ is $45 °$ .
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 $\overrightarrow {{\rm{OA}}}$ .
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_7b:
Particle $P_{2}$ also moves in a straight line. The position of $P_{2}$ is given by $r = (\begin{matrix} - 1 \\ 6 \end{matrix}) + t (\begin{matrix} 4 \\ - 3 \end{matrix})$ .

The speed of $P_{1}$ is greater than the speed of $P_{2}$ when $t > q$ .

Find the value of $q$ .
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 $\overrightarrow {{\text{AC}}}$ .
SPM.1.AHL.TZ0.8:
The plane П has the Cartesian equation $2x + y + 2z = 3$

The line L has the vector equation r $= \left( {\begin{array}{*{20}{c}} 3 \\ { - 5} \\ 1 \end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}} 1 \\ { - 2} \\ p \end{array}} \right){\text{,}}\,\,\mu {\text{,}}\,p \in \mathbb{R}$ . The acute angle between the line L and the plane П is 30°.

Find the possible values of $p$ .
16N.1.AHL.TZ0.H_8b:
determine the coordinates of the point of intersection P.
17M.2.AHL.TZ2.H_9b:
Determine whether or not the lines (OA) and (BC) intersect.
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
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 ${\text{CDBA}}$ .
17M.1.SL.TZ2.S_9b.i:
Find $\overrightarrow {{\text{AB}}}$ ;
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 $L$ .
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 $\overrightarrow {AB}$ .
21M.1.AHL.TZ1.11a.ii:
Find a vector equation of $L_{1}$ .
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 $y$ .
17N.1.AHL.TZ0.H_2b:
Find the coordinates of the point of intersection of the line L with the plane Π.
19N.2.SL.TZ0.S_2a:
Find the point of intersection of ${L_1}$ and ${L_2}$ .
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 $q$ .
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.
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.
17M.2.AHL.TZ2.H_9g:
Find the acute angle between the planes Π $_2$ and Π $_3$ .
SPM.2.AHL.TZ0.7:
Two ships, A and B , are observed from an origin O. Relative to O, their position vectors at time t hours after midday are given by

r_A = $\left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 5 \\ 8 \end{array}} \right)$

r_B = $\left( {\begin{array}{*{20}{c}} 7 \\ { - 3} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 0 \\ {12} \end{array}} \right)$

where distances are measured in kilometres.

Find the minimum distance between the two ships.
18M.1.SL.TZ1.S_9b.ii:
Point C (k , 12 , −k) is on L. Show that k = 14.
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.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.
20N.1.SL.TZ0.S_7a:
Find an expression for the velocity of $P_{1}$ at time $t$ .
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_2a:
Find a vector equation of the line L passing through the points A and B.
18N.1.AHL.TZ0.H_9c:
Show that L does not intersect the $y$ -axis for any negative value of $b$ .
17M.2.AHL.TZ2.H_9e:
Verify that 2j + k is perpendicular to the plane Π $_2$ .
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 $\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to$ .
18N.1.AHL.TZ0.H_9a:
Find, in terms of $b$ , a Cartesian equation of the plane Π containing this triangle.
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 $C$ .
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 $p = 2$ .
19M.1.SL.TZ1.S_2a:
Find $c$ .
EXN.2.AHL.TZ0.11e:
Find the volume of right-pyramid $ABCD$ .
17M.1.SL.TZ2.S_9b.ii:
Find $\left| {\overrightarrow {{\text{AB}}} } \right|$ .
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 $r = (\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}) + λ (\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix})$ .

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Topic 3— Geometry and trigonometry » AHL 3.15—Classification of lines

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