A B C D E F G H is a regular octagon. A B → = p , A H → = q , H G → = r and H C → = s . Using the vectors p , q , r and s , write down expressions for E B → .

Given that: A B → = p , A H → = q , H G → = r and H C → = s . We have to write down expressions for B C → . B C → is the resultant vector formed from A B → + A H → + H C → So, B C → = A B → + A H → + H C → ⇒ B C → = A H → + H C → - B A → ⇒ B C → = q + s - p Hence, B C → = q + s - p Now, E B → is the resultant vector formed from B C → + C D → + D E → So, E B → = B C → + C D → + D E → ⇒ E B → = q + s - p + r + q ⇒ E B → = 2 q + s + r - p Hence, E B → = 2 q + s + r - p .

A B C D E F G H is a regular octagon. A B → = p , A H → = q , H G → = r and H C → = s . Using the vectors p , q , r and s , write down expressions for E B → .

Given that: A B → = p , A H → = q , H G → = r and H C → = s . We have to write down expressions for B C → . B C → is the resultant vector formed from A B → + A H → + H C → So, B C → = A B → + A H → + H C → ⇒ B C → = A H → + H C → - B A → ⇒ B C → = q + s - p Hence, B C → = q + s - p Now, E B → is the resultant vector formed from B C → + C D → + D E → So, E B → = B C → + C D → + D E → ⇒ E B → = q + s - p + r + q ⇒ E B → = 2 q + s + r - p Hence, E B → = 2 q + s + r - p .

Relative to an origin O , the position vectors of the points A and B are given by O A → = 5 i + 3 j and O B → = 5 i + λ j + μ k In the case where λ = 0 and μ = 7 , find the unit vector in the direction of A B → .

Given that: O A → = 5 i + 3 j and O B → = 5 i + λ j + μ k We have to find the unit vector in the direction of A B → In the case where λ = 0 and μ = 7 , O B → = 5 i + 0 j + 7 k As we know that, A B → = O B → - O A → So, A B → = 5 - 5 i + 0 - 3 j + 7 - 0 k = - 3 j + 7 k Now, A B ^ = A B → A B → So, the unit vector in the direction of A B → = - 3 j + 7 k 3 2 + 7 2 = 1 58 - 3 j + 7 k .

Relative to an origin O , the position vectors of the points A and B are given by O A → = 5 i + 3 j and O B → = 5 i + λ j + μ k Given that triangle O A B is right-angled with area 5 34 and μ = 10 , find the value of λ .

Given that: O A → = 5 i + 3 j and O B → = 5 i + λ j + μ k and triangle O A B is right-angled with area 5 34 and μ = 10 We have to find the value of λ In the case where μ = 7 , O B → = 5 i + λ j + 10 k As we know that, Area of △ O A B = 1 2 O B → × O A → O A → × O B → = i j k 5 3 0 5 λ 10 = 30 i - 50 j + 5 λ - 15 k O A → × O B → = 30 2 + 50 2 + 5 λ - 15 2 But, ar △ O A B = 5 34 (given) So, 1 2 30 2 + 50 2 + 5 λ - 15 2 = 5 34 Squaring on both sides 1 4 30 2 + 50 2 + 5 λ - 15 2 = 25 × 34 ⇒ 30 2 + 50 2 + 5 λ - 15 2 = 25 × 34 × 4 ⇒ 5 λ - 15 2 = 0 ⇒ λ = 3 So, the required value of λ = 3

The position vectors of the points A , B , C and D are given by a , b , c and d , respectively, where a = 2 2 0 , b = 13 5 4 , c = 5 - 3 4 and d = - 6 - 6 0 . Find | A D → | , | B C → | , | A B → | and | D C → | .

Given that: The position vectors of the points A , B , C and D are given by a , b , c and d , respectively, where a = 2 2 0 , b = 13 5 4 , c = 5 - 3 4 and d = - 6 - 6 0 . We have to find | A D → | , | B C → | , | A B → | and | D C → | . | A D → | = d - a = - 6 - 6 0 - 2 2 0 = - 8 - 8 0 | A D → | = 128 = 8 2 | A B → | = b - a = 13 5 4 - 2 2 0 = 11 3 4 | A B → | = 146 | B C → | = c - b = 5 - 3 4 - 13 5 4 = - 8 - 8 0 | B C → | = 8 2 | B C → | = c - d = 5 - 3 4 - - 6 - 6 0 = 11 3 4 | B C → | = 146 .

HARD

AS and A Level

IMPORTANT

Earn 100

$A B C D E F G H$ is a regular octagon. $\vec{A B} = p, \vec{A H} = q, \vec{H G} = r$ and $\vec{H C} = s .$

Using the vectors $p, q, r$ and $s,$ write down expressions for $\vec{E B}$ .

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 9A>Q 6

$A B C D E F G H$ is a regular octagon. $\vec{A B} = p, \vec{A H} = q, \vec{H G} = r$ , $\vec{H C} = s$

Question Image

and $s = (1 + \sqrt{k}) p$ then find the exact value of $k .$

MEDIUM

AS and A Level

IMPORTANT

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

Use a vector method to show that, when the diagonals of a quadrilateral bisect one another, then the opposite sides are parallel and equal in length.

MEDIUM

AS and A Level

IMPORTANT

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

Relative to an origin $O,$ the position vectors of the points $A$ and $B$ are given by $\vec{O A} = 5 i + 3 j$ and $\vec{O B} = 5 i + λ j + μ k$

In the case where $λ = 0$ and $μ = 7,$ find the unit vector in the direction of $\vec{A B} .$

HARD

AS and A Level

IMPORTANT

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

Relative to an origin $O,$ the position vectors of the points $A$ and $B$ are given by $\vec{O A} = 5 i + 3 j$ and $\vec{O B} = 5 i + λ j + μ k$

Given that triangle $O A B$ is right-angled with area $5 \sqrt{34}$ and $μ = 10,$ find the value of $λ .$

MEDIUM

AS and A Level

IMPORTANT

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

The diagram shows a cuboid $O A B C D E F G$ with a horizontal base $O A B C .$ The cuboid has a length $O A$ of $8 cm,$ a width $A B$ of $4 cm$ and a height $B F$ of $d cm .$ The point $M$ is the midpoint of $C E$ and the point $N$ is the point on $G F$ such that $G N = 6 cm .$ The unit vectors $i, j$ and $k$ are parallel to $O A, O C$ and $O D,$ respectively. It is given that $\vec{O M} = 4 i + 2 j + k .$

Question Image

Write down the height of the cuboid. (Enter the answer excluding units)

HARD

AS and A Level

IMPORTANT

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

Question Image

Find $\vec{O N} .$

HARD

AS and A Level

IMPORTANT

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

Question Image

Find the unit vector in the direction $\vec{M N} .$

MEDIUM

AS and A Level

IMPORTANT

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

The position vectors of the points $A, B, C$ and $D$ are given by $a, b, c$ and $d,$ respectively, where

$a = (\begin{array}{l} 2 \\ 2 \\ 0 \end{array}), b = (\begin{matrix} 13 \\ 5 \\ 4 \end{matrix}), c = (\begin{array}{r} 5 \\ - 3 \\ 4 \end{array})$ and $d = (\begin{array}{r} - 6 \\ - 6 \\ 0 \end{array}) .$

Find $| \vec{A D} |, | \vec{B C} |, | \vec{A B} |$ and $| \vec{D C} | .$

ABCDEFGH is a regular octagon. AB→=p, AH→=q, HG→=r and HC→=s. Using the vectors p, q, r and s, write down expressions for EB→.

Important Questions on Vectors

Learn and Prepare for any exam you want !

$A B C D E F G H$ is a regular octagon. $\vec{A B} = p, \vec{A H} = q, \vec{H G} = r$ and $\vec{H C} = s .$

Using the vectors $p, q, r$ and $s,$ write down expressions for $\vec{E B}$ .