The points O , A , B , X and Y are such that O A → = a → , O B → = b → , O X → = 3 a → and O Y → = 3 b → . Find B X → and A Y → in terms of a → and b → . Further, if the point P divides A Y → in the ratio 1 : 3 then express B P → in terms of a → and b → .

Given, O A → = a → , O B → = b → , O X → = 3 a → and O Y → = 3 b → B X → = O X → - O B → = 3 a → - b → A Y → = O Y → - O A → = 3 b → - a → The position vector of point R which divides the line joining two points, P = a → and Q = b → in the ratio m : n internally is given by m b → + n a → m + n . Point P divides A Y → in the ratio 1 : 3 , ⇒ O P → = 1 × O Y → + 3 O A → 4 ⇒ O P → = 3 b → + 3 a → 4 Now, B P → = O P → - O B → = 3 b → + 3 a → 4 - b → = 3 b → + 3 a → - 4 b → 4 ∴ B P → = 1 4 ( 3 a → - b → )

Find the vector equation of the line passing through the point 2 i ^ + j ^ + 3 k ^ and parallel to the vector - 4 i ^ + 3 j ^ - k ^ .

Let, a → = 2 i ^ + j ^ + 3 k ^ and b → = - 4 i ^ + 3 j ^ - k ^ The vector equation of the line passing through a → and parallel to b → is r → = a → + t b → , t ∈ R r → = 2 i ^ + j ^ + 3 k ^ + t ( - 4 i ^ + 3 j ^ - k ^ ) , t ∈ R ∴ The vector equation is r ¯ = ( 2 - 4 t ) i ^ + ( 1 + 3 t ) j ^ + ( 3 - t ) k ^ , t ∈ R .

O A B C is a parallelogram. If O A → = a → and O C → = c → , find the vector equation of the side B C .

Let O be the origin. Given, O A → = a → and O C → = b → The side B C is parallel to O A → i.e., a → and passing through C is c → ∴ The vector equation is r → = c → + t a → where t ∈ R .

EASY

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IMPORTANT

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If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors, then test for the collinearity of the following points whose position vectors are given by $3 \vec{a} - 4 \vec{b} + 3 \vec{c}, - 4 \vec{a} + 5 \vec{b} - 6 \vec{c}, 4 \vec{a} - 7 \vec{b} + 6 \vec{c}$ .

Important Questions on Addition of Vectors

EASY

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 4 - Addition of Vectors>Exercise 4 (a)>Q II

\vec{a}, \vec{b}, \vec{c}

are non-coplanar vectors, then test for the collinearity of the following points whose position vectors are given by

2 \vec{a} + 5 \vec{b} - 4 \vec{c}, \vec{a} + 4 \vec{b} - 3 \vec{c}, 4 \vec{a} + 7 \vec{b} - 6 \vec{c}

EASY

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 4 - Addition of Vectors>Exercise 4 (a)>Q III

In the Cartesian plane,

O

is the origin of the coordinate axes. A person starts at

O

and walks a distance of

3

units in the NORTH-EAST direction and reaches the point

P

. From

P

he walks

4

units of distance parallel to NORTH-WEST direction and reaches the point

Q

. Express the vector

\vec{O Q}

in terms of

\hat{i}

and

\hat{j}

(

observe that

∠ E O P = 45 °)

MEDIUM

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 4 - Addition of Vectors>Exercise 4 (a)>Q III

The points

O, A, B, X

and

Y

are such that

\vec{O A} = \vec{a}, \vec{O B} = \vec{b}, \vec{O X} = 3 \vec{a}

and

\vec{O Y} = 3 \vec{b}

. Find

\vec{B X}

and

\vec{A Y}

in terms of

\vec{a}

and

\vec{b}

. Further, if the point

P

divides

\vec{A Y}

in the ratio

1 : 3

then express

\vec{B P}

in terms of

\vec{a}

and

\vec{b}

MEDIUM

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 4 - Addition of Vectors>Exercise 4 (a)>Q III

Δ O A B, E

is the midpoint of

A B

and

F

is a point on

O A

such that

O F = 2 F A

. If

C

is the point of intersection of

O E

and

B F

, then find the ratios

O C : C E

and

B C : C F

MEDIUM

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 4 - Addition of Vectors>Exercise 4 (a)>Q III

The point

E

divides the segment

P Q

internally in the ratio

1 : 2

and

R

is any point not on the line

P Q

. If

F

is a point on

Q R

such that

Q F : F R = 2 : 1,

then show that

E F

is parallel to

P R

EASY

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 4 - Addition of Vectors>Exercise 4 (b)>Q I

Find the vector equation of the line passing through the point

2 \hat{i} + \hat{j} + 3 \hat{k}

and parallel to the vector

- 4 \hat{i} + 3 \hat{j} - \hat{k}

EASY

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 4 - Addition of Vectors>Exercise 4 (b)>Q I

O A B C

is a parallelogram. If

\vec{O A} = \vec{a}

and

\vec{O C} = \vec{c}

, find the vector equation of the side

B C

EASY

11th Telangana Board

IMPORTANT

Intermediate First Year Mathematics paper - 1A>Chapter 4 - Addition of Vectors>Exercise 4 (b)>Q I

\vec{a}, \vec{b}, \vec{c}

are the position vectors of the vertices

A, B

and

C

respectively of

Δ A B C,

then find the vector equation of the median through the vertex

A

If a→, b→, c→ are non-coplanar vectors, then test for the collinearity of the following points whose position vectors are given by 3a→-4b→+3c→, -4a→+5b→-6c→, 4a→-7b→+6c→.

Important Questions on Addition of Vectors

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If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors, then test for the collinearity of the following points whose position vectors are given by $3 \vec{a} - 4 \vec{b} + 3 \vec{c}, - 4 \vec{a} + 5 \vec{b} - 6 \vec{c}, 4 \vec{a} - 7 \vec{b} + 6 \vec{c}$ .