The vector 1 3 2 i ^ - 2 j ^ + k ^ is-

perpendicular to the vector 3 i ^ + 2 j ^ - 2 k ^

makes an angle π 3 with vector 2 i ^ - 4 j ^ + 3 k ^

Faces A B C and B C D of a tetrahedron A B C D meet at an angle of 30 ° . The area of face A B C is 120 sq. units and the area of face B C D is 80 sq. units and B C = 10 units, then the volume of tetrahedron is

Given: ar ∆ A B C = 120 ⇒ a → × c → = 240 ar ∆ B C D = 80 ⇒ c → × d → = 160 c → = 10 Now, a → × c → × c → × d → = a → c → d → c → - a → c → c → d → ⇒ a → × c → × c → × d → = a → c → d → c → ⇒ a → × c → c → × d → sin 30 ° = a → c → d → c → ⇒ 240 × 160 × 1 2 = a → c → d → · 10 ⇒ a → c → d → = 1920 Volume of tetrahedron = 1 6 a → c → d → = 1920 6 = 320 unit

In ∆ A B C , Let B 3 , 3 , - 9 and A B = A C and altitude A D is x - 1 1 = y - 2 2 = z + 3 3 . If coordinates of C are α , β , γ , then α + β + γ is

We have, Equation of A D is x - 1 1 = y - 2 2 = z + 3 3 = λ say Then, D ≡ λ + 1 , 2 λ + 2 , 3 λ - 3 B D → = λ - 2 i ^ + 2 λ - 1 j ^ + 3 λ + 6 k ^ Now, B D ⊥ A D , therefore B D → · i ^ + 2 j ^ + 3 k ^ = 0 ⇒ λ - 2 + 2 2 λ - 1 + 3 3 λ + 6 = 0 ⇒ 14 λ + 14 = 0 ⇒ λ = - 1 Hence, D ≡ 0 , 0 , - 6 and B 3 , 3 , - 9 Since, D is mid-point of B C , therefore let C ≡ α , β , γ , then α + 3 2 = 0 ⇒ α = - 3 β + 3 2 = 0 ⇒ β = - 3 γ - 9 2 = - 6 ⇒ γ = - 3 ∴ α + β + γ ≡ 9

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If $\vec{a}, \vec{b}, \vec{c}, \vec{d}, \vec{e}, \vec{f}$ are position vectors of $6$ points $A, B, C, D, E$ and $F$ respectively such that $3 \vec{a} + 4 \vec{b} = 6 \vec{c} + \vec{d} = 4 \vec{e} + 3 \vec{f} = \vec{x}$ , then

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Important Questions on Vector Algebra

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-2>Q 24

The vector

\frac{1}{3} (2 \hat{i} - 2 \hat{j} + \hat{k})

is-

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-2>Q 25

If a vector

\vec{r}

of magnitude

3 \sqrt{6}

is collinear with the bisector of the angle between the vector

\vec{a} = 7 \hat{i} - 4 \hat{j} - 4 \hat{k}

and

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

, then

\vec{r} =

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-3>Q 2

Faces

A B C

and

B C D

of a tetrahedron

A B C D

meet at an angle of

30 °

. The area of face

A B C

120

sq. units and the area of face

B C D

80

sq. units and

B C = 10

units, then the volume of tetrahedron is

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-3>Q 3

∆ A B C

, Let

B (3, 3, - 9)

and

A B = A C

and altitude

A D

\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{3}

. If coordinates of

C

are

(α, β, γ)

, then

|α| + |β| + |γ|

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-3>Q 13

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

and

\vec{d}

are the position vectors of the points

A \equiv (x, y, z); B \equiv (y, - 2 z, 3 x); C \equiv (2 z, 3 x, - y)

and

D \equiv (1, - 1, 2)

respectively. If

|\vec{a}| = 2 \sqrt{3}; (\vec{a}^\vec{b}) = (\vec{a}^\vec{c}); (\vec{a}^\vec{d}) = \frac{π}{2}

and

(\vec{a}^\vec{j})

is obtuse, then the value of

(x + y + z)

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-3>Q 14

Let

\vec{u}

be a vector on rectangular coordinate system with sloping angle

60 °

. Suppose that,

|\vec{u} - \hat{i}|

is geometric mean of

|\vec{u}|

and

|\vec{u} - 2 \hat{i}|

where

\hat{i}

is the unit vector along

x

-axis then

|\vec{u}|

has the value equal to

\sqrt{a} - \sqrt{b}

where

a, b \in N .

Then the value of

{(a + b)}^{3} + {(a - b)}^{3}

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-3>Q 15

Let

\vec{A} = 2 \vec{i} + \vec{k}, \vec{B} = \vec{i} + \vec{j} + \vec{k}

and

\vec{C} = 4 \vec{i} - 3 \vec{j} + 7 \vec{k}

, then the vector

\vec{R}

, satisfying

\vec{R} \times \vec{B} = \vec{C} \times \vec{B}

and

\vec{R} \cdot \vec{A} = 0,

x \hat{i} + y \hat{j} + z \hat{k}

, then

(x + y + z)

is equal to

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-3>Q 16

A B C D

is a tetrahedron with p.v.'s of its angular point as

A (- 5, 22, 5); B (1, 2, 3); C (4, 3, 2)

and

D (- 1, 2, - 3) .

If the area of the triangle

A E F

where the quadrilaterals

A B D E

and

A B C F

are parallelograms is

\sqrt{S},

then the value of

S

If a→, b→, c→, d→, e→, f→ are position vectors of 6 points A, B, C, D, E and F respectively such that 3a→+4b→=6c→+d→=4e→+3f→=x→, then

Important Questions on Vector Algebra

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If $\vec{a}, \vec{b}, \vec{c}, \vec{d}, \vec{e}, \vec{f}$ are position vectors of $6$ points $A, B, C, D, E$ and $F$ respectively such that $3 \vec{a} + 4 \vec{b} = 6 \vec{c} + \vec{d} = 4 \vec{e} + 3 \vec{f} = \vec{x}$ , then