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

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Let $A$ and $B$ be two points in space with position vectors $\vec{a}$ and $\vec{b}$ respectively, then the real number $k$ such that the system of equations $|3 \vec{r} - 2 \vec{a} - \vec{b}| = |\vec{a} - \vec{b}|$ and $\{\vec{r} - k \vec{a} - (1 - k) \vec{b}\} \cdot (\vec{a} - \vec{b}) = 0$ does not have any solution, is

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Important Questions on Vector Algebra

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 7

Let the vectors $\vec{P Q}, \vec{Q R}, \vec{R S}, \vec{S T}, \vec{T U}$ and $\vec{U P}$ represent the sides of a regular hexagon.

STATEMENT- $1 :$ $\vec{P Q} \times (\vec{R S} + \vec{S T}) \neq \vec{0} .$

STATEMENT- $2 :$ $\vec{P Q} \times \vec{R S} = \vec{0}$ and $\vec{P Q} \times \vec{S T} \neq \vec{0} .$

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

The number of distinct real values of

λ

, for which the vectors

- λ^{2} \hat{i} + \hat{j} + \hat{k}, \hat{i} - λ^{2} \hat{j} + \hat{k}

and

\hat{i} + \hat{j} - λ^{2} \hat{k}

are coplanar, is

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 2

Let

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

be unit vectors such that

\vec{a} + \vec{b} + \vec{c} = 0

. Which one of the following is correct

?

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 3

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors

\hat{a}, \hat{b}, \hat{c}

such that

\hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{c} = \hat{c} \cdot \hat{a} = \frac{1}{2}

. Then the volume of the parallelopiped is

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 4

Let two non-collinear unit vectors

\hat{a}

and

\hat{b}

form an acute angle. A point

P

moves, so that at any time

t

the position vector

O P

(where,

O

is the origin) is given by

\hat{a} \cos t + \hat{b} \sin t

. When

P

is farthest from origin

O

, let

M

be the length of

\vec{O P}

and

\hat{u}

be the unit vector along

\vec{O P}

. Then,

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 5

Let

P (3, 2, 6)

be a point in space and

Q

be a point on the line

\vec{r} = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (- 3 \hat{i} + \hat{j} + 5 \hat{k}) .

Then the value of

μ

for which the vector

\vec{P Q}

is parallel to the plane

x - 4 y + 3 z = 1

is

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 6

If

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

and

\vec{d}

are the unit vectors such that

(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 1

and

\vec{a} \cdot \vec{c} = \frac{1}{2}

, then

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 7

Let

P, Q, R

and

S

be the points on the plane with position vectors

- 2 \hat{i} - \hat{j}, 4 \hat{i}, 3 \hat{i} + 3 \hat{j}

and

- 3 \hat{j} + 2 \hat{j}

, respectively. The quadrilateral

P Q R S

must be a