Find two unit vectors each of which is perpendicular to both u ¯ and v ¯ , where u ¯ = 2 i ^ + j ^ - 2 k ^ , v ¯ = i ^ + 2 j ^ - 2 k ^

Given, u ¯ = 2 i ^ + j ^ - 2 k ^ , v ¯ = i ^ + 2 j ^ - 2 k ^ Vector Perpendicular to the both u ¯ and v ¯ is u ¯ × v ¯ . So, u ¯ × v ¯ = i ^ j ^ k ^ 2 1 - 2 1 2 - 2 = i ^ ( - 2 + 4 ) - j ^ ( - 4 + 2 ) + k ^ ( 4 - 1 ) = 2 i ^ + 2 j ^ + 3 k ^ Unit vector in the direction of u ¯ × v ¯ is: ± u ¯ × v ¯ u ¯ × v ¯ = ± 2 i ^ + 2 j ^ + 3 k ^ 2 2 + 2 2 + 3 2 = ± 2 i ^ + 2 j ^ + 3 k ^ 17 = ± 2 17 i ^ + 2 17 j ^ + 3 17 k ^

Find the values of c so that for all real x the vectors x c i ^ - 6 j ^ + 3 k ^ and x i ^ + 2 j ^ + 2 c x k ^ make an obtuse angle.

Let a ¯ = x c i ^ - 6 j ^ + 3 k ^ and b = x i ^ + 2 j ^ + 2 c x k ^ Consider a ¯ . b ¯ = x c i ^ - 6 j ^ + 3 k ^ . x i ^ + 2 j ^ + 2 c x k ^ = x c x + - 6 2 + 3 2 c x = c x 2 - 12 + 6 c x If angle between a ¯ and b ¯ is obtuse, a ¯ . b ¯ < 0 ∴ c x 2 - 12 + 6 c x < 0 ∴ c x 2 + 6 x < 12 ∴ c < 12 x 2 + 6 x ∴ c < 12 x 2 + 6 x + 9 - 9 = 12 x + 3 2 - 9 ∴ c < min 12 x + 3 2 - 9 Now, 12 x + 3 2 - 9 is minimum when x + 3 2 - 9 is maximum. i.e. x + 3 2 - 9 = ∞ - 9 = ∞ ∴ c < min 12 ∞ = 0 ∴ c < 0 Hence, the angle between a ¯ and b ¯ is obtuse if c < 0 .

Show that the sum of the length of projections of p i ^ + q j ^ + r k ^ on the coordinate axes, where p = 2 , q = 3 and r = 4 , is 9 .

Let a ¯ = p i ^ + q j ^ + r k ^ Projection of a ¯ on X - axis: = a ¯ . i ^ i ^ = p i ^ + q j ^ + r k ^ . i ^ 1 = p = 2 Similarly, projections of a on Y , Z axes are 3 and 4 respectively. ∴ Sum of these projections = 2 + 3 + 4 = 9

Determine whether a ¯ and b ¯ are orthogonal, parallel or neither. a ¯ = - 9 i ^ + 6 j ^ + 15 k ^ , b ¯ = 6 i ^ - 4 j ^ - 10 k ^ .

Given, a ¯ = - 9 i ^ + 6 j ^ + 15 k ^ = - 3 3 i ^ - 2 j ^ - 5 k ^ = - 3 2 6 i ^ - 4 j ^ - 10 k ^ = - 3 2 b ¯ ∴ a ¯ = - 3 2 b ¯ i.e. a ¯ is non-zero scaler multiple of b ¯ . Hence, a ¯ is parallel to b ¯ .

Determine whether a ¯ and b ¯ are orthogonal, parallel or neither. a ¯ = 2 i ^ + 3 j ^ - k ^ , b ¯ = 5 i ^ - 2 j ^ + 4 k ^

Given, a ¯ = 2 i ^ + 3 j ^ - k ^ , b ¯ = 5 i ^ - 2 j ^ + 4 k ^ Now, a . b = 2 i ^ + 3 j ^ - k ^ . 5 i ^ - 2 j ^ + 4 k ^ = 2 5 + 3 - 2 + - 1 4 = 10 - 6 - 4 = 0 Since, a ¯ and b ¯ are non-zero vectors and a . b = 0 Therefore, a ¯ is orthogonal to b ¯ .

Determine whether a ¯ and b ¯ are orthogonal, parallel or neither. a ¯ = - 3 5 i ^ + 1 2 j ^ + 1 3 k ^ , b ¯ = 5 i ^ + 4 j ^ + 3 k ^

Given, a ¯ = - 3 5 i ^ + 1 2 j ^ + 1 3 k ^ , b ¯ = 5 i ^ + 4 j ^ + 3 k ^ Now, a . b = - 3 5 i ^ + 1 2 j ^ + 1 3 k ^ . 5 i ^ + 4 j ^ + 3 k ^ = - 3 5 5 + 1 2 4 + 1 3 3 = - 3 + 2 + 1 = 0 Since, a ¯ and b ¯ are non-zero vectors and a . b = 0 Therefore, a ¯ is orthogonal to b ¯ .

If a line makes angles 90 ° , 135 ° , 45 ° with X , Y and Z axes respectively, then find its direction cosines.

Direction cosines of a line making angle α with X - axis, angle β with Y - axis and angle γ with Z - axis are l , m , n , where l = cos α , m = cos β , n = cos γ . Here, α = 90 ° , β = 135 ° , γ = 45 ° So, direction cosines are l = cos 90 ° = 0 m = cos 135 ° = cos 180 ° - 45 ° = - cos 45 ° = - 1 2 n = cos 45 ° = 1 2 ∴ Direction cosines are l , m , n = 0 , - 1 2 , 1 2 .

E M B I B E

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.2>Q 12

EASY

12th Maharashtra Board

IMPORTANT

Earn 100

Find the centroid of tetrahedron with vertices $K (5, - 7, 0), L (1, 5, 3), M (4, - 6, 3), N (6, - 4, 2)$ ?

Important Questions on Vectors

MEDIUM

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.3>Q 1

Find two unit vectors each of which is perpendicular to both

\bar{u}

and

\bar{v}

, where

\bar{u} = 2 \hat{i} + \hat{j} - 2 \hat{k}, \bar{v} = \hat{i} + 2 \hat{j} - 2 \hat{k}

MEDIUM

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.3>Q 2

If

\bar{a}

and

\bar{b}

are two vectors perpendicular to each other, prove that

(\bar{a} + \bar{b})^{2} = (\bar{a} - \bar{b})^{2}

HARD

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.3>Q 3

Find the values of

c

so that for all real

x

the vectors

x c \hat{i} - 6 \hat{j} + 3 \hat{k}

and

x \hat{i} + 2 \hat{j} + 2 c x \hat{k}

make an obtuse angle.

EASY

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.3>Q 4

Show that the sum of the length of projections of

p \hat{i} + q \hat{j} + r \hat{k}

on the coordinate axes, where

p = 2, q = 3

and

r = 4

, is

9

.

MEDIUM

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.3>Q 5

Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.

EASY

12th Maharashtra Board

IMPORTANT

Mathematics & Statistics (Arts & Science) Part 1 Standard 12>Chapter 5 - Vectors>Exercise 5.3>Q 6

Determine whether