3.10 Vector Equations of Lines

1 2a 2b 3a 3b 4 5a 5b 5c 6a 6b 7a 7b 7c 8a 8b 9a 9b

Question 1

Marks: 7

The line $l$ has equation $r = (\begin{matrix} 4 \\ 0 \\ 3 \end{matrix}) + λ (\begin{matrix} - 1 \\ - 2 \\ 5 \end{matrix})$ and point A has coordinates $(3, t, 2)$ . Given that the shortest distance between point A and the line is $\frac{\sqrt{645}}{15}$ units, find $t$ , where $t \in ℤ$ .

Key Concepts

Equation of a Line in Vector Form
Shortest Distance Between a Point and a Line

Question 2a

Marks: 6

A line $l_{1}$ has the equation $r_{1} = (2 + λ) i + (6 λ - 3) j + (5 + 2 λ) k$ and intersects the line $l_{2}$ with equation $r_{2} = 5 i + (7 - 4 μ) j + (- 3 - 7 μ) k$ at point P, when $λ = 3$ .

A third line $l_{3}$ runs parallel to $l_{1}$ and also intersects $l_{2}$ at point $X (t, t - 2, - 2 t)$ .

(a)

Find the parametric equations of

l_{3}

Key Concepts

Coincident, Parallel, Intersecting & Skew Lines
Equation of a Line in Parametric Form

Question 2b

Marks: 2

(b)

Find the distance

|PX|

Key Concepts

Magnitude of a Vector

Question 3a

Marks: 4

Consider the two intersecting lines $l_{1}$ and $l_{2}$ defined by the equations:

$l_{1} : r = (\begin{matrix} 9 \\ 18 \\ 11 \end{matrix}) + λ (\begin{matrix} - 6 \\ - 3 \\ k \end{matrix})$

$l_{2} : \frac{x + 5}{2} = \frac{y + t}{- 4} = \frac{z - 20}{3}$

(a)

Given that the angle between

l_{1}

and

l_{2}

1.281

rad, correct to 4 significant figures, find the value of

k

, where

k \in ℤ

Key Concepts

Angle Between Two Lines

Question 3b

Marks: 3

(b)

Find the value of

t

, giving your answer correct to 3 significant figures.

Key Concepts

Coincident, Parallel, Intersecting & Skew Lines

Question 4

Marks: 8

Consider the two lines $l_{1}$ and $l_{2}$ , where $l_{1}$ passes through the points $A (11, - 2, 3)$ and $B (4, 4, - 5)$ and $l_{2}$ is defined by the Cartesian equations $\frac{x + 7}{3} = \frac{2 y + 9}{6} = \frac{z + 4}{- 4}$

Find the shortest distance between the two lines.

Key Concepts

Equation of a Line in Cartesian Form
Shortest Distance Between Two Lines

Question 5a

Marks: 6

Consider the line $l_{1}$ as defined by the equation $r_{1} = (\begin{matrix} - 2 \\ 5 \\ - 8 \end{matrix}) + α (\begin{matrix} 2 \\ - 1 \\ 3 \end{matrix})$ .

A point $P (r, t, - r)$ lies at a distance of $\sqrt{405}$ units perpendicular from a point $X (17, 15, - 8)$ on $l_{1}$ .

(a)

Find all possible coordinates of

P

Key Concepts

Perpendicular Vectors
Magnitude of a Vector

Question 5b

Marks: 2

(b)

Given that

t > 0

, write down the set of parametric equations that defines the line

l_{2}

that passes through points

P

and

X

Key Concepts

Equation of a Line in Parametric Form

Question 5c

Marks: 3

A third line $l_{3}$ is defined by the equations $\frac{- x - 13}{5} = \frac{- y - 9}{4} = \frac{z - 4}{2}$ .

(c)

Determine the relationship between lines

l_{2}

and

l_{3}

Key Concepts

Coincident, Parallel, Intersecting & Skew Lines

Question 6a

Marks: 2

A wheelchair ramp is required to provide access to a building with a door that is located 22 cm above ground level. The maximum angle that a ramp must be from the horizontal is 4.8°.

(a)

Calculate the minimum horizontal distance that the ramp must extend out.

Key Concepts

Right-Angled Trigonometry

Question 6b

Marks: 8

The wheelchair ramp is supported by a steel frame. A cross section of the ramp can be seen in the diagram below. A metal strut joins M, the midpoint of [AC], to a point X on the line [AB]. [AB].XM=11.1 cm and $M \hat{X} C$ =90°.

q6a_3-10_vector-equations-of-lines_very-hard_ib_aa_hl_maths-diagram

(b)

Using the horizontal distance found in part (a) and assuming that point A is at the origin, use a vector method to calculate the length XB.

Key Concepts

Perpendicular Vectors

Question 7a

Marks: 2

Two drones X and Y are being flown over an area of rainforest to look for signs of illegal logging. Their positions relative to the observation centre, are given by

$r_{x} = (\begin{matrix} - 3 \\ 1.6 \\ 2.5 \end{matrix}) + t (\begin{matrix} 2 \\ - 2 \\ 1 \end{matrix})$ and $r_{y} = (\begin{matrix} 2.5 \\ 0 \\ - 2 \end{matrix}) + t (\begin{matrix} 1.5 \\ 6 \\ 4 \end{matrix})$

at time $t$ minutes after take-off, $0 \leq t < 20$ . All distances are in metres.

(a)

Verify that the two drones will not collide.

Key Concepts

Coincident, Parallel, Intersecting & Skew Lines

Question 7b

Marks: 6

(b)

Find the shortest distance between the two drones and the time at which it occurs.

Key Concepts

Basic Coordinate Geometry
Modelling with Differentiation

Question 7c

Marks: 6

A third drone Z begins its flight at $t = 8$ and its position relative to the observation centre is given by $r_{z} = (\begin{matrix} 2 \\ 1.5 \\ 4.5 \end{matrix}) + t (\begin{matrix} 3 \\ 4 \\ 1 \end{matrix})$

Each drone can observe a circular area of ground, $A,$ such that $A = 1.8 h^{2}$ where $h$ is the height of the drone above the ground in metres.

(c)

Show that the area of ground that can be observed by drone Z five minutes after it takes off overlaps with the area of ground that can be observed by drone Y at that time.

Key Concepts

Equation of a Line in Vector Form
Magnitude of a Vector

Question 8a

Marks: 5

Consider the tetrahedron ABCD, where A(3, 5, 8), B(-2, 3, 2), C(5, -1, 3) and D(-3, 0, 1). M is the midpoint of the line BC and point P lies along the line DM.

(a)

Given that the volume of the tetrahedron ABCP is

\frac{1}{3}

of the volume of the tetrahedron ABCD, find the Cartesian equations of the line going through points A and P.

Key Concepts

Displacement Vectors
Equation of a Line in Cartesian Form

Question 8b

Marks: 5

(b) X is the midpoint of [AD].

Find the coordinates of the point of intersection between the line found in part (a) and the line going through [MX].

Key Concepts

Coincident, Parallel, Intersecting & Skew Lines

Question 9a

Marks: 6

A car is moving at a constant speed of 15 ms^-1 in the direction parallel to the vector $3 i - 6 j .$ Two birds are perched at points $A (17, 28, 16)$ and $B (- 48, 128, 26)$ .

At $t = 0$ , the car is located at $(2, 4, 0)$ and the bird at point A starts to fly at a constant velocity of $\frac{7 \sqrt{365}}{10}$ ms^-1. The bird at point B begins to fly at a constant velocity in the direction of the vector $52 i - 60 j - 9 k$ when $t = 1.2$ .

When bird A reaches the position of $(44, - 24, 4)$ , both birds and the car lie in a straight line.

(a)

Find the equation of the line along which the birds and car lie.

Key Concepts

Kinematics using Vectors
Equation of a Line in Vector Form

Question 9b

Marks: 6

(b)

Find the speed at which bird B is travelling.

Key Concepts

Kinematics using Vectors

DP IB Maths: AA HL

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3.10 Vector Equations of Lines

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