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

The sides of parallelogram are 2 i ^ + 4 j ^ - 5 k ^ and i ^ + 2 j ^ + 3 k ^ . Find the unit vectors, parallel to their diagonals.

Here, D A → = a → = 2 i ^ + 4 j ^ - 5 k ^ , D B → = b → = i ^ + 2 j ^ + 3 k ^ By, parallelogram law of vector addition, D C → = D A → + D B → = a → + b → = 3 i ^ + 6 j ^ - 2 k ^ . Then D C ^ = 3 i ^ + 6 j ^ - 2 k ^ 3 2 + 6 2 + - 2 2 = 3 7 i ^ + 6 7 j ^ - 2 7 k ^ . Now, D B → + B A → = D A → ⇒ B A → = D A → - D B → = a → - b → = i ^ + 2 j ^ - 8 k ^ Then B A ^ = i ^ + 2 j ^ - 8 k ^ 1 2 + 2 2 + - 8 2 = 1 69 i ^ + 2 69 j ^ - 8 69 k ^ So, the required vectors parallel to the diagonals are 3 7 i ^ + 6 7 j ^ - 2 7 k ^ and 1 69 i ^ + 2 69 j ^ - 8 69 k ^ .

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If $(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$ , where $\vec{a}, \vec{b}$ and $\vec{c}$ are any three vectors such that $\vec{a} \cdot \vec{b} \neq 0, \vec{b} \cdot \vec{c} \neq 0$ then $\vec{a}$ and $\vec{c}$ are

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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 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 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

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-4>Q 1

Let

A B C D

be a tetrahedron such that the edges

A B, A C

and

A D

are mutually perpendicular. Let the area of triangles

A B C, A C D

and

A D B

be denoted by

x, y

and

z

sq. units respectively. Find the area of the triangle

B C D

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Beta Question Bank for Engineering: Mathematics>Chapter 21 - Vector Algebra>EXERCISE-4>Q 2

The sides of parallelogram are

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

and

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

. Find the unit vectors, parallel to their diagonals.

If a→×b→×c→=a→×b→×c→, where a→, b→ and c→ are any three vectors such that a→·b→≠0, b→·c→≠0 then a→ and c→ are

Important Questions on Vector Algebra

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If $(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$ , where $\vec{a}, \vec{b}$ and $\vec{c}$ are any three vectors such that $\vec{a} \cdot \vec{b} \neq 0, \vec{b} \cdot \vec{c} \neq 0$ then $\vec{a}$ and $\vec{c}$ are