Vector Equations of Lines (DP IB Applications & Interpretation (AI): HL): Exam Questions

Point $A$ has coordinates $\left(7,-1,20\right)$  and the line $l$  is defined by the equations:

$l:\{\begin{array}{l}x=3+\lambda \\ y=2\lambda -1\\ z=\lambda \end{array}$

Point $\mathrm{B}$ lies on the line $l$ such that $\left[\mathrm{AB}\right]$ is perpendicular to $l$.

Find the coordinates of point $\mathrm{B}$.

Hence find the shortest distance from A to the line $l$.

Find the vector equation of the line $l_1$ with Parametric equations

$l_1:\{\begin{array}{l}x=4\lambda -3\\ y=2+5\lambda \\ z=4\lambda +3\end{array}$ 

A second line $l_2$  runs parallel to $l_1$ and passes through the points $\mathrm{X}(t,2,-3)$ and $\mathrm{Y}(23,22,q)$.

Find the value of $t$ and $q$ .

Hence write down the equation of line $l_2$ in Parametric form.

A line $l$ passes through the points $\mathrm{P}(6,5,-2)$ and $\mathrm{Q}\left(2x+2,x-5,x\right)$ and lies perpendicular to the vector $3i+4j-k$.

Find the vector equation of  $l$.

Find the obtuse angle formed by the two lines $l_1$ and $l_2$ defined by the equations:

$l_1:\{\begin{array}{l}x=4-2\lambda \\ y=1+5\lambda \\ z=\lambda -1\end{array}$

$l_2:\{\begin{array}{l}x=4+3\mu \\ y=18+\mu \\ z=6+2\mu \end{array}$

Consider the two lines $l_1$ and $l_2$ as defined by:

$l_1:\{\begin{array}{l}x=5+\mu \\ y=3-\mu \\ z=2\mu -8\end{array}$

$l_2:r=\left(\begin{array}{c}-4\\ 3\\ 1\end{array}\right)+\lambda \left(\begin{array}{c}2\\ -5\\ 2\end{array}\right)$

Find a vector that is perpendicular to both lines.

Hence find the shortest distance between the two lines.

Consider the lines  $l_1$ and $l_2$ defined by the equations:

 $l_1:\{\begin{array}{l}x=2+6\lambda \\ y=2+q\lambda \\ z=-8-5\lambda \end{array}$

$l_2:r=\left(\begin{array}{c}-4\\ 5\\ p\end{array}\right)+\lambda \left(\begin{array}{c}-24\\ 12\\ 20\end{array}\right)$ 

Given that  $l_1$ and $l_2$ are identical, find the value of $p$ and $q$.

Consider the two lines $l_1$ and $l_2$ defined by the equations:

$l_1:r=\left(\begin{array}{c}4\\ 4\\ -3\end{array}\right)+\lambda \left(\begin{array}{c}2\\ 1\\ -2\end{array}\right)$

$l_2:r=\left(\begin{array}{c}3\\ -1\\ -2\end{array}\right)+\mu \left(\begin{array}{c}-1\\ 1\\ 4\end{array}\right)$.

Show that the lines are not parallel and do not intersect.

Calculate the exact value of the acute angle between the lines.

A helicopter is hovering in the sky at coordinates (4.5, 8, 2.7) relative to a helipad positioned on the ground at the origin, O.

The $x$ direction is due east, the $y$ direction is due north and the $z$ direction is vertically upwards. The distances are measured in kilometres.

Write down the equation of a line the helicopter should travel along for it to travel directly to the helipad.

Assuming the helicopter travels directly towards the helipad, but stops to hover at a point, $P,$ 0.54 km vertically above the ground, find

i) the coordinates of the point $P$,

ii) the distance the helicopter has left to travel on its final descent to the helipad, given that it continues along the most direct route.

Assuming instead the helicopter travels directly towards a point, $Q$, 0.04 km vertically above the helipad, and then descends vertically downwards to the ground, find

i) The coordinates of the point $Q$,

ii) The component of the direction the helicopter actually travelled in that is perpendicular to the direction vector found in part (a),

iii) The distance the helicopter has travelled from its starting position, including the vertical descent from $Q$ to the helipad.

Consider the triangle ABC. The points A, B and C have coordinates $\left(-6,3,13\right),\left(4,5,-8\right)$  and $\left(3,-4,t\right)$  respectively.  A vector equation of the line that passes through point $\mathrm{A}$ and the midpoint of $\left[\mathrm{BC}\right]$  is $r=\left(\begin{array}{c}-6\\ 3\\ 13\end{array}\right)+\lambda \left(\begin{array}{c}19\\ -5\\ -27\end{array}\right)$ 

Find the value of $t$.

Find the vector equation of the line that passes through point B and the midpoint of [AC].

The two lines intersect inside the triangle at point X.

Show that the area of $\mathrm{AXC}$ is  $\frac{1}{3}$ the area of triangle $\mathrm{ABC}$ .

In the magical kingdom of Cartesia, all positions are measured relative to the ancient stone of power known as the Origin. This reference system corresponds to the standard $x,y,z$ coordinate system used in mathematics, as shown in the diagram below.

Prince Vector, son of the King Prime of Cartesia, needs to fly on his magical unicorn from the top of the Mystic Pedestal all the way to Cloud City, on an urgent rescue mission. 

The Mystic Pedestal is 14 kilometres west and 8 kilometres north of the Origin, and its top is one kilometre up from the level of the Origin. Cloud City is 11 kilometres east and 13 kilometres north of the Origin, and it is 11 kilometres up from the level of the Origin. 

Since there is not much time, the prince must fly directly from the top of the Mystic Pedestal to Cloud City. Unfortunately, the unicorn’s magic levels are low. In order for the unicorn to recharge it must pass within 12 kilometres of the Origin during the flight, and must do this before reaching the halfway point between the Mystic Pedestal and Cloud City. If the unicorn does not recharge before this point then it and the prince will crash into the barren wastes and the kingdom will perish. 

Using a vector method, determine whether or not the prince will reach Cloud City successfully. Use clear mathematical workings to justify your answer.

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

A line $l_1$ has the equation $r_1=\left(2+\lambda \right)\mathrm{i}+\left(6\lambda -3\right)\mathrm{j}+\left(5+2\lambda \right)k$ and intersects the line $l_2$ with equation $r_2=5\mathrm{i}+\left(7-4\mu \right)\mathrm{j}+\left(-3-7\mu \right)k$ at point P, when $\lambda =3$.

A third line $l_3$ runs parallel to $l_1$ and also intersects $l_2$ at point $X\left(t,t-2,-2t\right)$.  

Find the parametric equations of $l_3$.

Find the distance $\left|\mathrm{PX}\right|$.

Consider the two intersecting lines $l_1$ and $l_2$ defined by the equations:

$l_1:r=\left(\begin{array}{c}9\\ 18\\ 11\end{array}\right)+\lambda \left(\begin{array}{c}-6\\ -3\\ k\end{array}\right)$

$l_2:\{\begin{array}{l}x=2\mu -5\\ y=-4\mu -t\\ z=3\mu +20\end{array}$

Given that the angle between $l_1$ and $l_2$  is 1.281 radians, correct to 4 significant figures, find the value of $k$, where $k\in \mathrm{\mathbb{Z}}$.

Find the value of $t$, giving your answer correct to 3 significant figures.

Consider the two lines $l_1$ and $l_2$ , where $l_1$ passes through the points $\mathrm{A}(11,-2,3)$  and $\mathrm{B}\left(4,4,-5\right)$  and $l_2$  is defined by the Parametric equations:  

 $l_2:\{\begin{array}{l}x=3\mu -7\\ 2y=6\mu -9\\ z=-4\mu -4\end{array}$ 

Find the shortest distance between the two lines.

Consider the line $l_1$ as defined by the equation $r_1=\left(\begin{array}{c}-2\\ 5\\ -8\end{array}\right)+\alpha \left(\begin{array}{c}2\\ -1\\ 3\end{array}\right)$. 

A point $\mathrm{P}(r,t,-r)$ lies at a distance of $\sqrt{405}$ units perpendicular from a point $\mathrm{X}\left(17,15,-8\right)$  on $l_1$.

Find all possible coordinates of P.

Given that $t>0$, write down the set of parametric equations that defines the line $l_2$ that passes through points P and X.

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°.

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

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 $\mathrm{M}\hat{\mathrm{X}}\mathrm{C}$=90°.  

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.

Some children are watching a canal boat navigating a system of locks. The boat starts at coordinates $\left(-10,-2,-7\right)$  relative to the point at which the children are standing.

The $x$direction is due east, the $y$ direction is due north and the $z$ direction is vertically upwards. All distances are measured in metres and the children are taken to be standing at the origin.

The boat travels with direction vector $1.5\mathrm{i}+2\mathrm{j}$ for 10 metres to get into the lock and then descends vertically downwards in the lock for 11 metres before continuing along the same direction vector as it was travelling along before entering the lock.

Find the coordinates of the entrance of the lock, given that the boat is now closer to the children.

Find the equation of the line along which the boat is travelling after it leaves the lock

On the next part of the journey at the point when the boat is closest to the children a child throws a flower to the boat driver. Given that the flower travels in a straight line and is caught by the boat driver, find the distance that the flower travelled.

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

Given that the volume of the tetrahedron ABCP is  $\frac{1}{3}$ of the volume of the tetrahedron ABCD, find the Vector equation of the line going through points A and P.

 X is the midpoint of $\left[\mathrm{AD}\right]$ .

Find the coordinates of the point of intersection between the line found in part (a) and the line going through $\left[\mathrm{MX}\right]$.

An adventure park structure is made out of steel rods arranged into a frame. As a part of the structure a red rod joins the coordinates $\left(2,-26,21\right)$ to $\left(-6,14,-23\right)$ and a blue rod joins $\left(16,-33,-46\right)$ to $\left(6,-18,-21\right)$.

Find the coordinates of the point where the red and blue rods meet each other.

The red rod also meets a yellow rod which has the vector equation $r=\left(s-1\right)\mathrm{i}+\left(s-29\right)\mathrm{j}+\left(8s-3\right)k$. The point intersection of the red and blue rods and the red and yellow rods are joined by a taut rope.

Find the length of the rope.

A graphics designer joins the coordinates $A\left(1,2,3\right)$ to $B\left(1,0,1\right)$ and also plots the line $l$ with parametric equations:

$l:\{\begin{array}{l}x=3-2\lambda \\ y=\lambda -6\\ z=1-\lambda \end{array}$

Find a Vector equation of the line joining the points $A$ and $B$ and show that it does not intersect the line $l$.

Find the two possible coordinates of the point $C$ on $l$ such that the angle $BAC$ is equal to  $\frac{\mathrm{\pi }}{3}$ radians.