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

The two adjacent sides of a parallelogram are 2 i ^ - 4 j ^ + 5 k ^ and i ^ - 2 j ^ - 3 k ^ . Find the unit vector parallel to its diagonal.

Draw a parallelogram and mark adjacent sides and also draw the diagonal. Since the two adjacent sides of a parallelogram are given in the question, its diagonal can be found out by adding the two adjacent sides (triangle law), d → = 2 i ^ - 4 j ^ + 5 k ^ + i ^ - 2 j ^ - 3 k ^ ⇒ d → = 3 i ^ - 6 j ^ + 2 k ^ Now the unit vector parallel to diagonal can be found using u d → = d → d → . u d → = 3 i ^ - 6 j ^ + 2 k ^ 3 2 + ( - 6 ) 2 + 2 2 ⇒ u d → = 3 i ^ - 6 j ^ + 2 k ^ 7 ⇒ u d → = 3 i ^ 7 - 6 j ^ 7 + 2 k ^ 7

The dot products of a vector with the vector ( i ^ + j ^ - 3 k ^ ) , ( i ^ + 3 j ^ - 2 k ^ ) and ( 2 i ^ + j ^ + 4 k ^ ) are 0 , 5 and 8 respectively. Find the vector.

Let the unknown vector be A → = a i ^ + b j ^ + c k ^ ∴ ( a i ^ + b j ^ + c k ^ ) · ( i ^ + j ^ - 3 k ^ ) = 0 ⇒ a + b - 3 c = 0 . . . . 1 ∴ ( a i ^ + b j ^ + c k ^ ) · ( i ^ + 3 j ^ - 2 k ^ ) = 5 ⇒ a + 3 b - 2 c = 5 . . . ( 2 ) ∴ ( a i ^ + b j ^ + c k ^ ) · ( 2 i ^ + j ^ + 4 k ^ ) = 8 ⇒ 2 a + b + 4 c = 8 . . . . . ( 3 ) Solving equations 1 , 2 , 3 , simultaneously we get: 1 - ( 2 ) ⇒ - 2 b - c = - 5 . . . . . 4 2 × 2 - 3 ⇒ 5 b - 8 c = 2 . . . . . ( 5 ) 5 × 4 + 2 × 5 ⇒ - 21 c = - 21 ⇒ c = 1 substituting c = 1 in equation ( 4 ) ⇒ - 2 b - 1 = - 5 ⇒ b = 2 substituting b , c values in equation ( 1 ) ⇒ a + 2 - 3 = 0 ⇒ a = 1 a = 1 , b = 2 , c = 1 ∴ A → = i ^ + 2 j ^ + k ^ .

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 →

Let u ^ = u 1 i ^ + u 2 j ^ + u 3 k ^ be a unit vector in R 3 and w ^ = 1 6 i ^ + j ^ + 2 k ^ . Given that there exists a vector v → in R 3 such that u ^ × v → = 1 and w ^ . u ^ × v → = 1 . Which of the following statement(s) is (are) correct ?

If u ^ lies in the xy - plane then u 1 = u 2

There is exactly one choice for such v →

If u ^ lies in the xz - plane then 2 u 1 = u 3

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 12

MEDIUM

JEE Main/Advance

IMPORTANT

Earn 100

Find the area of a parallelogram, if the diagonals of a parallelogram are represented by the vectors ${\vec{a}}_{1} = 3 \hat{i} + \hat{j} - 2 \hat{k}$ and ${\vec{a}}_{2} = \hat{i} - 3 \hat{j} + 4 \hat{k}$

Important Questions on Vector Algebra

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 12

If

| \vec{a} | = \sqrt{26}, | \vec{b} | = 7

and

|\vec{a} \times \vec{b}| = 35

, then find the value of

\vec{a} \cdot \vec{b} .

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 13

If the vectors

\vec{a} = 3 \hat{i} + \hat{j} - 2 \hat{k}

and

\vec{b} = \hat{i} + λ \hat{j} - 3 \hat{k}

are perpendicular to each other, then find the value of

λ .

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 13

Find the area of a triangle, if two adjacent sides of a triangle are represented by the vectors

\vec{a} = 3 \hat{i} + 4 \hat{j}

and

\vec{b} = - 5 \hat{i} + 7 \hat{j} .

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 14

The two adjacent sides of a parallelogram are

2 \hat{i} - 4 \hat{j} + 5 \hat{k} and \hat{i} - 2 \hat{j} - 3 \hat{k}

. Find the unit vector parallel to its diagonal.

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 15

Let

A

and

B

are two points with position vectors

\vec{a}

and

\vec{b}

respectively, and let

P

be a point dividing

A B

internally in the ratio

m : n

. Let

\vec{O P} = \vec{r}

, then prove that

\vec{r} = \frac{m \vec{b} + n \vec{a}}{m + n} .

HARD

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 16

Find a vector whose magnitude is $3$ units and which is perpendicular to the following two vectors $\vec{a} = 3 \hat{i} + \hat{j} - 4 \hat{k}, \vec{b} = 6 \hat{i} + 5 \hat{j} - 2 \hat{k}$ .

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment Set-1>Q 17

The dot products of a vector with the vector $(\hat{i} + \hat{j} - 3 \hat{k}), (\hat{i} + 3 \hat{j} - 2 \hat{k})$ and $(2 \hat{i} + \hat{j} + 4 \hat{k})$ are $0, 5$ and $8$ respectively. Find the vector.