Vector Equations of Lines & The Scalar Product (Edexcel IAL Maths): Exam Questions

The equation of a line is given in vector form as follows $r = 2 i + 6 j + t ( 7 i + 3 j ).$

Explain briefly what the vectors $( 2 i + 6 j )$and $( 7 i + 3 j )$ in the equation above tell us, respectively, about the line.

Write down, in vector form, the equation of the line passing through $( 1 , 4 )$ in the same direction as $(i\mathrm{+3}j).$

A line passes through the two points $A ( 3 , - 2 )$ and $B ( 5 , 7 )$.

(i) Write down the position vectors $\overset{\rightarrow}{OA}$and $\overset{\rightarrow}{OB}$ of the two points.

(ii) Use the vector relation $\overset{\rightarrow}{AB}=\overset{\rightarrow}{OB}-\overset{\rightarrow}{OA}$ to find vector $\overset{\rightarrow}{AB}$.

(iii) Use your answers from (i) and (ii) to write down an equation of the line in vector form.

Write down in vector form the equation of the line through the point $( 1 , 3 , - 6 )$ in the direction $4i-j+2k$.

By first calculating the vectors $\overset{\rightarrow}{OA},\overset{\rightarrow}{OB}$ and $\overset{\rightarrow}{AB}$, find a vector equation of the line passing through the two points $A ( 1 , 3 , 1 )$ and $B ( 3 , 4 , - 3 )$

$A$ and $B$ are the points on the line $r = ( \begin{gathered} 2 \\ - 1 \\ 3 \end{gathered} ) + t ( \begin{gathered} 0 \\ 4 \\ - 3 \end{gathered} )$with $t=1$ and $t=3$ respectively.

(i) Find the position vectors $\overset{\rightarrow}{OA}$and $\overset{\rightarrow}{OB}$.

(ii) By first finding the vector $\overset{\rightarrow}{AB}$, calculate the modulus $\left| \vec{AB} \right|$.

The coordinates of three points are $A ( 2 , - 1 , 3 )$, $B ( 1 , - 2 , 5 )$and $C ( 6 , 3 , 1 )$.

Find $\overset{\rightarrow}{AB}$ and $\overset{\rightarrow}{AC}$, and calculate $\left| \vec{AB} \right|$ and $\left| \vec{AC} \right|$ .

For two vectors $a=a_{1}i+a_{2}j+a_{3}k$ and $b=b_{1}i+b_{2}j+b_{3}k$, the scalar product $a.b$ can be calculated using the formula

$a.b=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$

Calculate the scalar product $\overset{\rightarrow}{AB.}\overset{\rightarrow}{AC}.$

A defining property of the scalar product of two vectors $a$ and $b$ is

$a . b = \text{stretchy} \mid a \text{stretchy} \mid \text{stretchy} \mid b \text{stretchy} \mid \text{cos} \theta$

where $θ$ is the angle between the two vectors.

Using this relationship, along with your answers to parts (a) and (b), find the angle between the vectors $\overset{\rightarrow}{AB}$ and $\overset{\rightarrow}{AC}$.

Give your answer in degrees correct to 1 decimal place.

For two vectors $a$ and $b$, the definition of the *scalar product * $a.b$ is

equation

Write down $\cos ( 90^{\circ} )$ , and explain why this value of cosine shows that if $a$ and $b$ are perpendicular then $a.b=0.$

For $a=a_{1}i+a_{2}j+a_{3}k$ and $b=b_{1}i+b_{2}j+b_{3}k$, the scalar product can be calculated using the formula

$a.b=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$

Additionally it can be shown that if $a.b=0$ then $a$ and $b$ are perpendicular.

Use the above results to prove that the vectors $\overset{\rightarrow}{AB}=-3i+j+5k$ and $\overset{\rightarrow}{AC}=i-7j+2k$are perpendicular.

Two lines $l$ and $m$ have equations $r = i + j + 3 k + s ( i - j + k )$ and $r = i - 2 j + 2 k + t ( 2 i + j + 3 k )$ respectively. The two lines intersect at a point $P$.

Explain why, at point $P$, the values of the parameters $s$ and $t$ must satisfy the vector equation

$( \begin{gathered} 1 + s \\ 1 - s \\ 3 + s \end{gathered} ) = ( \begin{gathered} 1 + 2 t \\ - 2 + t \\ 2 + 3 t \end{gathered} )$

Solve the simultaneous equations $1+s=1+2t$and $1-s=-2+t,$and show that the values found for $s$ and $t$also satisfy the equation $3+s=2+3t.$

Hence, find the coordinates of the point $P$.

Two lines $l$ and $m$ have equations $r = 6 i - 2 j + k + s ( i + j - k )$ and $r = 3 i + 3 j - 2 k + \mathrm{t} ( 3 i - j + k )$ respectively.

Show that if the two lines intersect, then at the point of intersection the parameters $s$ and $t$ must satisfy the vector equation

$( \begin{gathered} 6 + s \\ - 2 + s \\ 1 - s \end{gathered} ) = ( \begin{gathered} 3 + 3 t \\ 3 - t \\ - 2 + t \end{gathered} )$

Solve the simultaneous equations $6+s=3+3t$ and $-2+s=3-t,$ and show that the values found for $s$ and $t$do not also satisfy the equation $1-s=-2+t.$

What do the results of part (b) tell you about the lines $l$ and $m$?

The line $l$ has equation $r = ( \begin{gathered} 2 \\ - 3 \\ 4 \end{gathered} ) + s ( \begin{gathered} 1 \\ 2 \\ - 1 \end{gathered} )$.

Point $N$ is the point on line $l$ such that the line connecting $N$ and the point $P ( 1 , 7 , 5 )$ is perpendicular to $l$.

Explain why $\vec{ON} = ( 2 + s

• 3 + 2 s 4 - s )$for some value of the parameter$s,$and use that expression for$\overset{\rightarrow}{ON}$to show that the vector$\overset{\rightarrow}{NP}$is given in terms of$s$ by

$\vec{NP} = ( - 1 - s 10 - 2 s 1 + s )$

Using the properties of the scalar product, explain why the value of $s$ corresponding to point $N$ must satisfy the equation

$( - 1 - s 10 - 2 s 1 + s ). ( 1 2

• 1 ) = 0$

By solving the equation in part (b), determine the coordinates of point $N$.

Given that the shortest distance from a point to a line is the perpendicular distance, find the shortest distance from point $P$ to the line $l$. You should give your answer as an exact value.

Find the equation of each of these lines in vector form.

The line joining $(4,1)$ to $(7,6)$.

The line passing through $(2,5)$ in the same direction as $2i+j$.

Find the equation of each of these lines in vector form.

(i) Through $(3,1,-2)$ in the direction $( \begin{gathered} 0 \\ 1 \\ 5 \end{gathered} )$

(ii) Through $(2,0,7)$ and $(5,2,-3)$.

Show that the point $(-4,-4,27)$ lies on the line found in part (a) (ii).

$A$ and $B$ are the points on the line $r = ( \begin{gathered} 3 \\ - 2 \\ 5 \end{gathered} ) + t ( \begin{gathered} 1 \\ 3 \\ - 1 \end{gathered} )$ with $t=4$ and $t=7$ respectively.

(i) Find the position vectors $\overset{\rightarrow}{OA}$ and $\overset{\rightarrow}{OB}$.

(ii) Find $\left| \vec{AB} \right|$.

The coordinates of three points are $A(1,0,-4),B(3,-1,6)$and $C(-2,5,7)$.

Find $\overset{\rightarrow}{AB}$ and $\overset{\rightarrow}{AC}$.

Calculate $\left| \vec{AB} \right|$ and $\left| \vec{AC} \right|$, and the scalar product $\overset{\rightarrow}{AB}\cdot\overset{\rightarrow}{AC}$.

Hence, find the angle between the vectors $\overset{\rightarrow}{AB}$ and $\overset{\rightarrow}{AC}$.

Give your answer in degrees correct to 1 decimal place.

The vertices of triangle $ABC$ are the points with coordinates $A(2,4,-3),B(0,2,1)$ and $C(-4,-2,-3)$.

(i) Show that $\overset{\rightarrow}{BA}\cdot\overset{\rightarrow}{BC}=0$.

(ii) What does the result of part (i) tell you about the triangle $ABC$?

Two lines $l$ and $m$ have equations $r=-3i-6j+8k+s(i+j-k)$ and $r=8i-10j+6k+t(2i-3j+k)$ respectively.

Show that the two lines meet and find the position vector of the point of intersection.

Two lines $l$ and $m$ have equations $r=8i-5j-6k+s(-i+2j+k)$ and $r=2i+2j-2k+t(2i+j+k)$ respectively.

(i) Show that the two lines do not intersect.

(ii) Are the two lines skew? Be sure to justify your answer.

The line $l$ has equation $r = ( \begin{gathered} 1 \\ - 4 \\ 10 \end{gathered} ) + s ( \begin{gathered} 1 \\ - 1 \\ 3 \end{gathered} ).$

Point $N$ is the point on line $l$ such that the line connecting $N$ and the point $P(3,4,1)$ is perpendicular to $l$.

Find the position vector, $\overset{\rightarrow}{ON}$, of point $N$.

Hence find the shortest distance from point $P$ to the line $l$.

The following diagram depicts a large bank vault, the walls, floor and ceiling of which are all perfectly rectangular.

The corner $O$ is taken to be the origin of the coordinate system, and the units on the diagram are all given in metres.

$CE$ is the diagonal running across the room from corner $C$ to corner $E$.

By first finding the vector , write a vector equation of the line that passes through and .

A motion detector is located at point $O$, and is set to sound an alarm if anything moves within 5 metres of it.

If a small insect flies in a straight line from point $C$ to point $E$, will the alarm be triggered? You must provide clear mathematical workings to justify your answer.

Find the equation of each of these lines in vector form.

The line joining $(3,-2)$ to $(-1,5)$.

The line passing through $(3,1)$ parallel to $4i-12j$.

Find the equation of each of these lines in vector form.

(i) Through $(2,-7,9)$ in the same direction as $6i+3j-9k$.

(ii) Through $(4,-3,1)$ and $(-8,3,-5)$.

Show that the point $(10,-6,5)$ does not lie on the line found in part (a) (ii).

$A$ and $B$ are the points on the line $r = ( \begin{gathered} 1 \\ - 4 \\ 3 \end{gathered} ) + t ( \begin{gathered} 2 \\ a \\ - 1 \end{gathered} )$ with $t=-1$ and $t=2$ respectively. Given that the distance from $A$ to $B$ is 9 units, find the possible values of $a$.

The coordinates of three points are $A(2,-1,5),B(4,1,1)$ and $C(-2,-5,9)$.

Find $\overset{\rightarrow}{BA}$ and $\overset{\rightarrow}{BC}$.

By considering the scalar product $\vec{BA} \times \vec{BC}$, or otherwise, calculate the angle between $\overset{\rightarrow}{BA}$ and $\overset{\rightarrow}{BC}$. Give your answer in degrees, accurate to 1 decimal place.

The vertices of triangle $ABC$ are the points with coordinates $A(5,-3,0),B(-2,0,-1)$ and $C(1,2,1)$.

(i) Calculate the scalar products $\vec{AB} \times \vec{AC} , \vec{BA} \times \vec{BC}$ and $\vec{CA} \times \vec{CB.}$

(ii) What does the result of part (i) tell you about the triangle $ABC$?

Determine whether each of the following pairs of lines intersect, are parallel, or are skew. If the lines intersect, find the coordinates of the point of intersection.

(i) $r=-7i-j+6k+s(i+j-k)$ and $r=2i+2j+11k+t(2i-j+5k)$

(ii) $r=3i-3j+k+s(i-2j+k)$ and $r=-i-j+5k+t(2i+2j-3k)$.

Find the perpendicular distance of the point $P(6,0,3)$ to the line $r=-i+5j+2k+s(3i-6j-2k)$.

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 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 to save the kingdom from certain doom.

The Mystic Pedestal is 14 miles west and 8 miles north of the Origin, and its top is one mile up from the level of the Origin. Cloud City is 11 miles east and 13 miles north of the Origin, and it is 11 miles 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, his unicorn’s magic levels are low. In order for the unicorn to recharge it must pass within 12 miles 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.

After the prince departs, his sister Hypatia (a keen mathematics student and heiress to the throne) remembers that there is a vector method that can be used to determine whether or not her brother’s mission will succeed.

Determine whether or not the prince will reach Cloud City successfully, using clear mathematical workings to justify your answer.

Given that the coordinates of $A$ and $B$ are $( - 1 , 7 )$ and $( - 7 , - 5 )$respectively, find the equation of each of the following lines in vector form.

(i) The line joining $( 5 , - 6 )$ to the midpoint of $AB$.

(ii) The line passing through $B$, parallel to the line in part (i).

Given that the coordinates of $A,B$ and $C$are $( 6 , 1 , - 3 ) , ( - 2 , - 7 , 5 )$ and $( 3 , 12 , - 9 )$ respectively, find the equation of each of the following lines in vector form.

(i) The line through $A$ and $B$.

(ii) The line through $B$, parallel to $\overset{\rightarrow}{OC}$.

Determine whether the point $( 1 , 5 , - 4 )$ lies on either of the lines found in part (a).

The coordinates of three points are $A ( - 2 , 3 , - 5 )$, $B ( 2 , - 5 , - 9 )$ and $C ( - 10 , - 1 , - 1 )$.

The point $M$ is the midpoint of $AB$, and the point $N$ lies on $BC$.

Given that $\left| \vec{BN} \right| = 3 \left| \vec{NC} \right|$, find the equation of the line through points $M$ and $N$ in vector form.

The coordinates of three points are A ( - 3 , 4 , 0 ) , B ( 2 , - 2 , - 3 ) and C ( - 1 , - 1 , - 4 ).

Calculate the angle between $\overset{\rightarrow}{CA}$ and $\overset{\rightarrow}{CB}$. Give your answer in degrees, accurate to 1 decimal place.

The vertices of triangle $ABC$are the points with coordinates A ( - 2 , 5 , 4 ) , B ( 3 , 1 , 0 ) and C ( - 1 , - 3 , - 1 ).

Use a vector method to prove that $ABC$ is a right-angled triangle.

Determine whether each of the following pairs of lines intersect, are parallel, or are skew. For any lines that intersect, determine the point(s) of intersection.

(i) $r = 3 i - 5 j + k + \mathrm{s} ( i + 3 j - 2 k )$ and r = i + 2 j + 2 k and t ( i - j - k ).

(ii) $r = i - 2 j + 3 k + \mathrm{s} ( 2 i + 6 j - 2 k )$ and $r = 4 i + 7 j + \mathrm{t} ( - 5 i - 15 j + 5 k )$

(iii) $r = - i - 4 j - 4 k + \mathrm{s} ( 2 i - 2 j - 4 k )$and $r = - 4 i - j - 2 k + \mathrm{t} ( 3 i - 3 j + 6 k )$

Find the coordinates of the point on the line $r = 2 i - 12 j + 3 k + s ( i - 6 j + 4 k )$ that is closest to the point $P ( 2 , 3 , - 1 ) ,$ and hence determine the minimum distance from point $P$ to the line.

The following diagram depicts imaginary lines connecting five points in space:

Points $A,B,C$ and $D$ are the locations, respectively, of the stars Arccirclus, Betacarotjuse, α-Capella and Denomineb. Point $S$ is the location of the Stellamortis battle station, a planet-killing atrocity being built by the evil Galactic Imperium. Coordinates are given relative to an origin point in accordance with the standard $x,y,z$ coordinate system, and the units for all coordinates are parsecs.

The forces of the Star Rebellion are prepared to launch a strike to destroy the battle station, but they are unsure of its exact location. According to data recovered from a smuggled droid, however, the following facts are known about the location of point $S$:

• Point $S$ is in the First Octant of the galaxy, where $x,y$ and $z$ coordinates are all positive.

• The distance from point $C$ to point $S$ is exactly $45\sqrt{2}$ parsecs.

• Points $B,C,D$ and $S$ form the base of a pyramid, with its apex at point $A$.

• The point on $BD$ closest to point $A$ is also the point where the two diagonals of the pyramid’s base intersect.

As the rebellion’s Chief Mathematician, it is your job to use the information provided to find the exact coordinates of point $S$. The fate of the galaxy is in your mathematical hands!