In the triangle P Q R , P Q → = 2 a ¯ and Q R → = 2 b ¯ . The mid-point of P R is M . Find P R → in terms of a ¯ and b ¯ .

P Q ¯ = 2 a ¯ , Q R ¯ = 2 b ¯ Using triangle law for addition of two vectors, P R ¯ = P Q ¯ + Q R ¯ = 2 a ¯ + 2 b ¯

In the triangle P Q R , P Q → = 2 a ¯ and Q R → = 2 b ¯ . The mid-point of P R is M . Find P M → in terms of a ¯ and b ¯ .

P Q ¯ = 2 a ¯ , Q R ¯ = 2 b ¯ Using triangle law for addition of two vectors, P R ¯ = P Q ¯ + Q R ¯ = 2 a ¯ + 2 b ¯ P M ¯ = 1 2 P R ¯ = 1 2 2 a ¯ + 2 b ¯ = a ¯ + b ¯

In the triangle P Q R , P Q → = 2 a ¯ and Q R → = 2 b ¯ . The mid-point of P R is M . Find Q M → in terms of a ¯ and b ¯ .

P Q ¯ = 2 a ¯ , Q R ¯ = 2 b ¯ Using triangle law for addition of two vectors, P R ¯ = P Q ¯ + Q R ¯ = 2 a ¯ + 2 b ¯ R M ¯ = 1 2 R P ¯ = - 1 2 P R ¯ = - 1 2 2 a ¯ + 2 b ¯ = - a ¯ - b ¯ Q M = Q R + R M = 2 b ¯ + - a ¯ - b ¯ = - a ¯ + b ¯ = b ¯ - a ¯

O A B C D E is a regular hexagon. The points A and B have position vectors a ¯ and b ¯ respectively, referred to the origin O . Find, in terms of a ¯ and b ¯ the position vectors of C , D and E .

Given : O A ¯ = a ¯ , O B ¯ = b ¯ . Let A D , B E , O C meet at M . Then M bisects A D , B E , O C . A B ¯ = A O ¯ + O B ¯ = - O A ¯ + O B ¯ = - a ¯ + b ¯ ∵ O A B M is a parallelogram. O M ¯ = A B ¯ = b ¯ - a ¯ O C ¯ = 2 O M ¯ = 2 b ¯ - a ¯ = 2 b ¯ - 2 a ¯ Now, O D ¯ = O C ¯ + C D ¯ = O C ¯ - D C ¯ = O C ¯ - O A ¯ ∵ O A = D C , O A ∥ D C = 2 b ¯ - 2 a ¯ - a ¯ = 2 b ¯ - 3 a ¯ And, O E = O M ¯ + M E ¯ = O M ¯ - E M ¯ = b ¯ - a ¯ - a ¯ ∵ E M ¯ = O A ¯ and E M ¯ ∥ O A ¯ = b ¯ - 2 a ¯ Hence, the position vectors of C , D and E are 2 b ¯ - 2 a ¯ , 2 b ¯ - 3 a ¯ and b ¯ - 2 a ¯ respectively.

If A B C D E F is a regular hexagon, show that A B ¯ + A C ¯ + A D ¯ + A E ¯ + A F ¯ = 6 A O ¯ , where O is the centre of the hexagon.

A B C D E F is a regular hexagon. ∴ A B ¯ = E D ¯ and A F ¯ = C D ¯ ∴ By the triangle law of addition of vectors, A C ¯ + A F ¯ = A C ¯ + C D ¯ = A D ¯ A E ¯ + A B ¯ = A E ¯ + E D ¯ = A D ¯ L.H.S. = A B ¯ + A C ¯ + A D ¯ + A E ¯ + A F ¯ = A D ¯ + A C ¯ + A F ¯ + A B ¯ + A E ¯ = A D ¯ + A D ¯ + A D ¯ = 3 A D ¯ = 3 2 A O ¯ ( ∵ O is the mid-point of A D ¯ ) = 6 A O ¯ Hence proved.

Check whether the vectors 2 i ^ + 2 j ^ + 3 k ^ , - 3 i ^ + 3 j ^ + 2 k ^ and 3 i ^ + 4 k ^ form a triangle or not.

Let, if possible, the three vectors form a triangle A B C . A B ¯ = 2 i ^ + 2 j ^ + 3 k ^ , B C = - 3 i ^ + 3 j ^ + 2 k ^ , A C = 3 i ^ + 4 k ^ Now, A B ¯ + B C = 2 i ^ + 2 j ^ + 3 k ^ + - 3 i ^ + 3 j ^ + 2 k ^ = 2 - 3 i ^ + 2 + 3 j ^ + 3 + 2 k ^ = - i ^ + 5 j ^ + 5 k ^ ≠ 3 i ^ + 4 k ^ = A C ¯ Hence, the three vectors do not form a triangle.

In the figure express c ¯ and d ¯ in terms of a ¯ and b ¯ . Find a vector in the direction of a ¯ = i ^ - 2 j ^ that has magnitude 7 units.

By triangle law, P Q ¯ = P S ¯ + S Q ¯ ∴ a ¯ = c ¯ - d ¯ … (1) Also, P R ¯ = P S ¯ + S R ¯ ∴ b ¯ = c ¯ + d ¯ … (2) Adding equations (1) and (2), we get ∴ a ¯ + b ¯ = c ¯ - d ¯ + c ¯ + d ¯ = 2 c ¯ ∴ c ¯ = 1 2 a ¯ + b ¯ = 1 2 a ¯ + 1 2 b ¯ Substituting for c ¯ in (2), we get d ¯ = b ¯ - 1 2 a ¯ - 1 2 b ¯ = 1 2 b ¯ - 1 2 a ¯ Hence, c ¯ = 1 2 a ¯ + 1 2 b ¯ , d ¯ = 1 2 b ¯ - 1 2 a ¯ Given vector is a ¯ = i ^ - 2 j ^ The unit vector in the direction of given vector a ¯ is a ^ = a → a → = i ^ - 2 j ^ 1 2 + - 2 2 = i ^ - 2 j ^ 5 Therefore, the vector in the direction of vector a → whose magnitude is 7 units is 7 5 i ^ - 2 j ^

E M B I B E

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.1

Maharashtra Board Solutions for Chapter: Vectors, Exercise 1: Exercise 5.1

Author:Maharashtra Board

Maharashtra Board Mathematics Solutions for Exercise - Maharashtra Board Solutions for Chapter: Vectors, Exercise 1: Exercise 5.1

Attempt the practice questions on Chapter 5: Vectors, Exercise 1: Exercise 5.1 with hints and solutions to strengthen your understanding. Mathematics & Statistics (Arts & Science) Part 1 Standard 12 solutions are prepared by Experienced Embibe Experts.

Questions from Maharashtra Board Solutions for Chapter: Vectors, Exercise 1: Exercise 5.1 with Hints & Solutions

EASY

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.1>Q 2

In the triangle $P Q R, \vec{P Q} = 2 \bar{a}$ and $\vec{Q R} = 2 \bar{b}$ . The mid-point of $P R$ is $M$ . Find $\vec{P R}$ in terms of $\bar{a}$ and $\bar{b}$ .

EASY

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.1>Q 2

In the triangle $P Q R, \vec{P Q} = 2 \bar{a}$ and $\vec{Q R} = 2 \bar{b}$ . The mid-point of $P R$ is $M$ . Find $\vec{P M}$ in terms of $\bar{a}$ and $\bar{b}$ .

EASY

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.1>Q 2

In the triangle $P Q R, \vec{P Q} = 2 \bar{a}$ and $\vec{Q R} = 2 \bar{b}$ . The mid-point of $P R$ is $M$ . Find $\vec{Q M}$ in terms of $\bar{a}$ and $\bar{b}$ .

HARD

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.1>Q 3

$O A B C D E$ is a regular hexagon. The points $A$ and $B$ have position vectors $\bar{a}$ and $\bar{b}$ respectively, referred to the origin $O$ . Find, in terms of $\bar{a}$ and $\bar{b}$ the position vectors of $C, D$ and $E$ .

EASY

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.1>Q 4

If $A B C D E F$ is a regular hexagon, show that $\bar{A B} + \bar{A C} + \bar{A D} + \bar{A E} + \bar{A F} = 6 \bar{A O}$ , where $O$ is the centre of the hexagon.

EASY

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.1>Q 5

Check whether the vectors $2 \hat{i} + 2 \hat{j} + 3 \hat{k}, - 3 \hat{i} + 3 \hat{j} + 2 \hat{k}$ and $3 \hat{i} + 4 \hat{k}$ form a triangle or not.

HARD

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.1>Q 6

In the figure express $\bar{c}$ and $\bar{d}$ in terms of $\bar{a}$ and $\bar{b}$ .
Find a vector in the direction of $\bar{a} = \hat{i} - 2 \hat{j}$ that has magnitude $7$ units.