Let a → = i ^ + 2 j ^ , b → = - 2 i ^ + j ^ and c → = 4 i ^ + 3 j ^ . Find scalars x and y such that c → = x a → + y b → .

Given, a → = i ^ + 2 j ^ , b → = - 2 i ^ + j ^ , c → = 4 i ^ + 3 j ^ Also given that c → = x a → + y b → ⇒ 4 i ^ + 3 j ^ = x i ^ + 2 j ^ + y - 2 i ^ + j ^ ⇒ 4 i ^ + 3 j ^ = x - 2 y i ^ + 2 x + y j ^ Two vectors are equal if their corresponding components are equal x - 2 y = 4 . . . . . . . i 2 x + y = 3 . . . . . . . i i Multiplying equation i i by 2 and adding to equation i , we get 5 x = 10 ⇒ x = 2 ∴ y = - 1

If a → = 2 i ^ + 2 j ^ - 3 k ^ and b → = 3 i ^ - j ^ + 2 k ^ . Find the angle between 2 a → + b → and a → + 2 b → .

Given a → = 2 i ^ + 2 j ^ - 3 k ^ and b → = 3 i ^ - j ^ + 2 k ^ Now, 2 a → + b → = 2 2 i ^ + 2 j ^ - 3 k ^ + 3 i ^ - j ^ + 2 k ^ = 4 i ^ + 4 j ^ - 6 k ^ + 3 i ^ - j ^ + 2 k ^ = 7 i ^ + 3 j ^ - 4 k ^ a → + 2 b → = 2 i ^ + 2 j ^ - 3 k ^ + 2 3 i ^ - j ^ + 2 k ^ = 2 i ^ + 2 j ^ - 3 k ^ + 6 i ^ - 2 j ^ + 4 k ^ = 8 i ^ + k ^ Let θ be the angle between 2 a → + b → and a → + 2 b → cos θ = 2 a → + b → · a → + 2 b → 2 a → + b → a → + 2 b → ⇒ cos θ = 7 i ^ + 3 j ^ - 4 k ^ · 8 i ^ + k ^ 7 i ^ + 3 j ^ - 4 k ^ 8 i ^ + k ^ ⇒ cos θ = 56 - 4 49 + 9 + 16 64 + 1 ⇒ cos θ = 52 74 65 ⇒ θ = cos - 1 52 74 65 .

If the position vectors of the points A ( 3 , 4 ) , B ( 5 , - 6 ) and C ( 4 , - 1 ) are a → , b → , c → respectively, evaluate a → + 2 b → - 3 c → .

Let a → , b → , c → are the position vectors of the points A ( 3 , 4 ) B ( 5 , - 6 ) and C ( 4 , - 1 ) Then, a → = 3 i ^ + 4 j ^ b → = 5 i ^ - 6 j ^ c → = 4 i ^ - j ^ Therefore, a → + 2 b → - 3 c → = 3 i ^ + 4 j ^ + 2 5 i ^ - 6 j ^ - 3 4 i ^ - j ^ = i ^ - 5 j ^

If a → , b → , c → are unit vectors such that a → + b → + c → = 0 → , find the value of a → ⋅ b → + b → ⋅ c → + c → ⋅ a → .

Given, a → , b → , and c → are unit vectors. So, a → = b → = c → = 1 ................ 1 Also, a → + b → + c → = 0 → So, a → + b → + c → = 0 → = 0 ............. 2 Now, a → + b → + c → 2 = a → + b → + c → ⋅ a → + b → + c → = a → · a → + a → · b → + a → · c → + b → · a → + b → · b → + b → · c → + c → · a → + c → · b → + c → · c → [Using distributivity of scalar product] Now, using the commutative property of dot product i.e. a → · b → = b → · a → = a → · a → + a → · b → + c → · a → + a → · b → + b → · b → + b → · c → + c → · a → + b → · c → + c → · c → = a → · a → + b → · b → + c → · c → + 2 a → · b → + b → · c → + c → · a → = a → 2 + b → 2 + c → 2 + 2 a → · b → + b → · c → + c → · a → [ Using a → · a → = a → 2 ] = 1 + 1 + 1 + 2 a → · b → + b → · c → + c → · a → [ Using 1 ] = 3 + 2 a → · b → + b → · c → + c → · a → ⇒ 0 = 3 + 2 a → · b → + b → · c → + c → · a → [Using 2 ] ⇒ a → ⋅ b → + b → ⋅ c → + c → ⋅ a → = - 3 2

If | a → | = 26 , | b → | = 7 and a → × b → = 35 , then find the value of a → · b → .

Given: a → × b → = 35 ⇒ a → b → sin θ = 35 where, θ is the angle between a → & b → . ⇒ 26 × 7 × sin θ = 35 ⇒ sin θ = 5 26 ⇒ cos θ = 1 26 Now, a → · b → = a → b → cos θ ⇒ a → · b → = 26 × 7 × 1 26 ⇒ a → · b → = 7

If the vectors a → = 3 i ^ + j ^ - 2 k ^ and b → = i ^ + λ j ^ - 3 k ^ are perpendicular to each other, then find the value of λ .

Given: a → = 3 i ^ + j ^ - 2 k ^ b → = i ^ + λ j ^ - 3 k ^ If a → is perpendicular to b → , then a → · b → = 0 ⇒ 3 i ^ + j ^ - 2 k ^ · i ^ + λ j ^ - 3 k ^ = 0 ⇒ 3 + λ + 6 = 0 ⇒ λ = - 9

Let a and b be positive real numbers. Suppose P Q → = a i ^ + b j ^ and P S → = a i ^ - b j ^ are adjacent sides of a parallelogram P Q R S . Let u → and v → be the projection vectors of w → = i ^ + j ^ along P Q → and P S → , respectively. If | u → | + v → = w → and if the area of the parallelogram P Q R S is 8 Sq. unit , then which of the following statements is/are TRUE?

The length of the diagonal P R → of the parallelogram P Q R S is 4 unit

w → is an angle bisector of the vectors P Q → and P S →

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 4

MEDIUM

JEE Main/Advance

IMPORTANT

Earn 100

Give a condition that three vectors form the three sides of a triangle. What are the other possibilities $?$

Important Questions on Vector Algebra

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JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 9

Let $\vec{a} = \hat{i} + 2 \hat{j}, \vec{b} = - 2 \hat{i} + \hat{j}$ and $\vec{c} = 4 \hat{i} + 3 \hat{j}$ . Find scalars $x$ and $y$ such that $\vec{c} = x \vec{a} + y \vec{b}$ .