The vector equation of the line L 1 is given by x y z = - 3 0 8 + t 4 - 1 - 3 . Show that A ( 5 , - 3 , 2 ) is the foot of the perpendicular from the point B ( 4 , - 7 , 2 ) to the line L 2 .

Given that: The vector equation of the line L 1 is given by x y z = - 3 0 8 + t 4 - 1 - 3 . We have to prove that A ( 5 , − 3 , 2 ) is the foot of the perpendicular from the point B ( 4 , − 7 , 2 ) to the line L 2 . We have the points A ( 5 , - 3 , 2 ) and B ( 4 , − 7 , 2 ) O A → = 5 - 3 2 , O B → = 4 - 7 2 B A → = O A → - O B → = 1 4 0 Now, B A → · d L 2 = 1 4 0 · 4 - 1 - 3 ⇒ B A → · d L 2 = 1 4 + 4 - 1 + 0 - 3 = 0 So, A ( 5 , − 3 , 2 ) is the foot of the perpendicular from the point B ( 4 , − 7 , 2 ) to the line L 2 . Hence, proved.

The points A , B and C are position vectors relative to the origin O , given by O A → = 3 1 5 , O B → = 0 - 1 2 and O C → = 1 2 3 . Find a vector equation for the line L passing through A and B .

Given that: O A → = 3 1 5 , O B → = 0 - 1 2 and O C → = 1 2 3 . We have to find a vector equation for the line L passing through A and B . As we know that, The vector equation for the line L passing through A and B is r = O B → + t A B → So, the required equation of line is r = O B → + t O B → - O A → r = 0 - 1 2 + t - 3 - 2 - 3 Or, r = - j + 2 k + t - 3 i - 2 j - 3 k

The points A , B and C are position vectors relative to the origin O , given by O A → = 3 1 5 , O B → = 0 - 1 2 and O C → = 1 2 3 . The line through C , perpendicular to L , meets L at the point N . Find the exact coordinates of N .

Given that: O A → = 3 1 5 , O B → = 0 - 1 2 and O C → = 1 2 3 . We have to find the exact coordinates of N . The line through C , perpendicular to L , meets L at the point N O N → = - 3 t + 3 - 2 t + 1 - 3 t + 5 and then C N → = - 3 t + 2 - 2 t - 1 - 3 t + 2 Since C N → is perpendicular to L So, - 3 t + 2 - 3 + - 2 t - 1 - 2 + - 3 t + 2 - 3 = 0 ⇒ 22 t - 10 = 0 ⇒ t = 5 11 O N → = 3 1 5 + 5 11 - 3 - 2 - 3 = 1 11 18 1 30 Hence, the exact coordinates of N 18 11 , 1 11 , 30 11 .

HARD

AS and A Level

IMPORTANT

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The line $L_{1}$ has vector equation $r = 2 i - 3 j + k + λ (6 i + j + 3 k)$ and the line $L_{2}$ has parametric equations $x = μ + 4, y = μ - 7, z = 3 μ .$

Show that $L_{1}$ and $L_{2}$ are not parallel.

Important Questions on Vectors

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9D>Q 6

The line $L_{1}$ has vector equation $r = 2 i - 3 j + k + λ (6 i + j + 3 k)$ and the line $L_{2}$ has parametric equations $x = μ + 4, y = μ - 7, z = 3 μ .$

Find the angle between the line $L_{1}$ and the line $L_{2} .$

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9D>Q 7

The vector equation of the line $L_{1}$ is given by $(\begin{array}{l} x \\ y \\ z \end{array}) = (\begin{array}{r} - 3 \\ 0 \\ 8 \end{array}) + t (\begin{array}{r} 4 \\ - 1 \\ - 3 \end{array}) .$

Find the vector equation of the line $L_{2}$ that is parallel to $L_{1}$ and which passes through the point $A (5, - 3, 2) .$

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9D>Q 7

The vector equation of the line $L_{1}$ is given by $(\begin{array}{l} x \\ y \\ z \end{array}) = (\begin{array}{r} - 3 \\ 0 \\ 8 \end{array}) + t (\begin{array}{r} 4 \\ - 1 \\ - 3 \end{array}) .$

Show that $A (5, - 3, 2)$ is the foot of the perpendicular from the point $B (4, - 7, 2)$ to the line $L_{2} .$

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9D>Q 8

The points $A, B$ and $C$ are position vectors relative to the origin $O,$ given by $\vec{O A} = (\begin{array}{l} 3 \\ 1 \\ 5 \end{array}), \vec{O B} = (\begin{array}{r} 0 \\ - 1 \\ 2 \end{array}) and \vec{O C} = (\begin{array}{l} 1 \\ 2 \\ 3 \end{array}) .$

Find a vector equation for the line $L$ passing through $A$ and $B .$

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9D>Q 8

The line through $C,$ perpendicular to $L,$ meets $L$ at the point $N .$ Find the exact coordinates of $N .$

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9E>Q 1

Find whether the following pairs of lines are parallel, intersecting or skew. State the position vector of the common point of any lines that intersect.

$x = 3 λ + 4 x = μ + 1$

$y = 5 λ + 1 y = μ + 4$

$z = 4 λ - 3 z = 2 μ + 5$

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9E>Q 1

Find whether the following pairs of lines are parallel, intersecting or skew. State the position vector of the common point of any lines that intersect.

$r = i - 5 j + 4 k + λ (3 i - 4 j + 9 k)$ $r = i - 5 j + 4 k + λ (- 6 i + 8 j - 18 k)$

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9E>Q 1

Find whether the following pairs of lines are parallel, intersecting or skew. State the position vector of the common point of any lines that intersect.

$r = 7 i + 2 j + 6 k + λ (i + 3 j + 9 k) r = 4 j + k + μ (8 i + j + 14 k)$

The line L1 has vector equation r=2i-3j+k+λ6i+j+3k and the line L2 has parametric equations x=μ+4, y=μ-7, z=3μ. Show that L1 and L2 are not parallel.

Important Questions on Vectors

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The line $L_{1}$ has vector equation $r = 2 i - 3 j + k + λ (6 i + j + 3 k)$ and the line $L_{2}$ has parametric equations $x = μ + 4, y = μ - 7, z = 3 μ .$

Show that $L_{1}$ and $L_{2}$ are not parallel.