Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-2>Q 24

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About the line $\frac{x - 1}{2} = \frac{y - 2}{- 3} = \frac{z + 3}{1}$ , the plane $3 x + 4 y + 6 z + 7 = 0$ is rotated till the plane passes through the origin. Now, $4 x + α y + β z = 0$ is the equation of plane in new position. The value of $α^{2} + β^{2}$ is

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Important Questions on Vector Algebra

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-2>Q 26

A vector

\vec{a}

has components

2 p

and

1

with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the counterclockwise sense. If with respect to the new system,

\vec{a}

has components

p + 1

and

1,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-2>Q 27

If

{\vec{z}}_{1} = a \vec{i} + b \hat{j}

and

{\vec{z}}_{2} = c \hat{i} + d \hat{j}

are two vectors in

\hat{i}

and

\hat{j}

system, where

|{\vec{z}}_{1}| = |{\vec{z}}_{2}| = r

and

{\vec{z}}_{1} \cdot {\vec{z}}_{2} = 0,

then

{\vec{w}}_{1} = a \vec{i} + c \vec{j}

and

{\vec{w}}_{2} = b \hat{i} + d \hat{j}

satisfy

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-2>Q 28

If

a, b, c, x, y, z \in R

such that

a x + b y + c z = 2,

then which of the following is always true

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-2>Q 29

The value

(s)

of

α \in [0, 2 π]

for which vector

\vec{a} = \hat{i} + 3 \hat{j} + (\sin 2 α) \hat{k}

makes an obtuse angle with the

z

-axis and the vectors

\vec{b} = (\tan α) \hat{i} - \hat{j} + 2 \sqrt{\sin \frac{α}{2} \hat{k}}

and

\vec{c} = (tanα) \hat{i} + (tanα) \hat{j} - 3 \sqrt{cosec \frac{α}{2}} \hat{k}

are orthogonal, is/are

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-2>Q 30

The vector

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

is rotated through an angle of

\cos^{- 1} \frac{11}{14}

and doubled in magnitude, then it becomes

4 \hat{i} + (4 x - 2) \hat{j} + 2 \hat{k} .

The value of

' x'

CANNOT be

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-2>Q 31

The vertices of a triangle are

A (1, 1, 2), B (4, 3, 1)

and

C (2, 3, 5) .

A vector representing the bisector of the angle

A

is:

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-2>Q 32

The vector

\vec{c},

parallel to the internal bisector of the angle between the vectors

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

and

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

with

|\vec{c}| = 5 \sqrt{6},

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-2>Q 34

\hat{a}

and

\hat{b}

are two given unit vectors at right angle. The unit vector equally inclined with

\hat{a}, \hat{b}

and

\hat{a} \times \hat{b}

will be