Let α ∈ R and the three vectors a → = α i ^ + j ^ + 3 k ^ , b → = 2 i ^ + j ^ - α k ^ and c → = α i ^ - 2 j ^ + 3 k ^ . Then the set S = { α : a → , b → and c → are coplanar}

contains exactly two positive numbers

contains exactly two numbers only one of which is positive

If a → , b → , c → are non-coplaner vectors such that b → × c → = a → ; c → × a → = b → ; a → × b → = c → , then which of the following is not TRUE?

| a → | = | b → | = | c → | = 2

| a → | | b → | | c → | = 1

EASY

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Let $\vec{u}$ and $\vec{v}$ be non-collinear vectors in $R^{2}$ . Let $\vec{w}$ be the orthogonal projection vector of $\vec{u}$ on $\vec{v}$ . Consider two statements:
(i) Any vector in $R^{2}$ can be written as a linear combination of $\vec{u}$ and $\vec{v}$
(ii) $\vec{w}$ can be written as a linear combination of $\vec{u}$ and $\vec{v}$ as $\vec{w} = a \vec{u} + b \vec{v}$ where both $a$ and $b$ are nonzero real numbers.

28.57% studentsanswered this correctly

Important Questions on Vector Algebra

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

\vec{a}

and

\vec{b}

be two unit vectors such that

\vec{a} . \vec{b} = 0 .

For some

x, y \in R,

let

\vec{c} = x \vec{a} + y \vec{b} + (\vec{a} \times \vec{b}) .

|\vec{c}| = 2

and the vector

\vec{c}

is inclined at the same angle

α

to both

\vec{a}

and

\vec{b},

then the value of

8 \cos^{2} α

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

P Q R S

be a quadrilateral. If

M

and

N

are midpoints of the sides

P Q

and

R S

respectively then

\vec{P S} + \vec{Q R} =

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

\vec{a} = \hat{i} + \hat{j} + 2 \hat{k}, \vec{b} = 2 \hat{i} - 3 \hat{j} + \hat{k}

and

\vec{c} = \hat{i} - \hat{j} + \hat{k}

be the three given vectors. Let

\vec{v}

be a vector in the plane of

\vec{a}

and

\vec{b}

whose projection on

\vec{c}

\frac{2}{\sqrt{3}}

. If

\vec{v}, \hat{j} = 7

, then

\vec{v} \cdot (\hat{i} + \hat{k})

is equal to

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

A, B, C, D

be the points with position vectors

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

and

4 \hat{i} + 5 \hat{j} + λ \hat{k}

respectively. If the points

A, B, C, D

lie on a plane, then the value of

λ

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

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

and

\vec{c} = \hat{i} - \hat{j} - \hat{k}

be three given vectors. If

\vec{r}

is a vector such that

\vec{r} \times \vec{a} = \vec{c} \times \vec{a}

and

\vec{r} \cdot \vec{b} = 0,

then

\vec{r} \cdot \vec{a}

is equal to

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

S

be the set of all real values of

λ

such that a plane passing through the points

(- λ^{2}, 1, 1), (1, - λ^{2}, 1)

and

(1, 1, - λ^{2})

also passes through the point

(- 1, - 1, 1)

. Then

S

is equal to :

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

\vec{a}

and

\vec{b}

be two unit vectors such that

|\vec{a} + \vec{b}| = \sqrt{3}

. If

\vec{c} = \vec{a} + 2 \vec{b} + (\vec{a} \times \vec{b})

, then

2 |\vec{c}|

is equal to:

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

The value of

m,

if the vectors

\hat{i} - \hat{j} - 6 \hat{k}, \hat{i} - 3 \hat{j} + 4 \hat{k}

and

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

are coplanar, is

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

\vec{x}

be a vector in the plane containing vectors

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

and

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

If the vector

\vec{x}

is perpendicular to

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

and its projection on

\vec{a}

\frac{17 \sqrt{6}}{2},

then the value of

{|\vec{x}|}^{2}

is equal to _______.

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

If $\vec{r} = 3 \hat{i} + 2 \hat{j} - 5 \hat{k}, \vec{a} = 2 \hat{i} - \hat{j} + \hat{k}, \vec{b} = \hat{i} + 3 \hat{j} - 2 \hat{k}$ and $\vec{c} = - 2 \hat{i} + \hat{j} - 3 \hat{k}$ such that $\vec{r} = λ \vec{a} + μ \vec{b} + γ \vec{c}$ , then

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

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

and

\vec{c} = x \hat{i} + (x - 2) \hat{j} - \hat{k}

and if the vector

\vec{c}

lies in the plane of vectors

\vec{a}

and

\vec{b},

then

x

equals

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

A vector

\vec{a} = α \hat{i} + 2 \hat{j} + β \hat{k} (α, β \in R)

lies in the plane of the vectors,

\vec{b} = \hat{i} + \hat{j}

and

\vec{c} = \hat{i} - \hat{j} + 4 \hat{k} .

\vec{a}

bisects the angle between

\vec{b}

and

\vec{c},

then

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

α \in R

and the three vectors

\vec{a} = α \hat{i} + \hat{j} + 3 \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - α \hat{k}

and

\vec{c} = α \hat{i} - 2 \hat{j} + 3 \hat{k} .

Then the set S = {

α : \vec{a}, \vec{b}

and

\vec{c}

are coplanar}

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

The sum of the distinct real values of

μ

for which the vectors

μ \hat{i} + \hat{j} + \hat{k}, \hat{i} + μ \hat{j} + \hat{k}, \hat{i} + \hat{j} + μ \hat{k}

are co-planar, is

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

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

are non-coplaner vectors such that

\vec{b} \times \vec{c} = \vec{a}; \vec{c} \times \vec{a} = \vec{b}; \vec{a} \times \vec{b} = \vec{c}

, then which of the following is not TRUE?

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

If the vectors

\vec{α} = \hat{i} + a \hat{j} + a^{2} \hat{k}, \vec{β} = \hat{i} + b \hat{j} + b^{2} \hat{k}

and

\vec{γ} = \hat{i} + c \hat{j} + c^{2} \hat{k}

are three non-coplanar vectors and

|\begin{array}{l} a & a^{2} & 1 + a^{3} \\ b & b^{2} & 1 + b^{3} \\ c & c^{2} & 1 + c^{3} \end{array}| = 0,

then the value of

a b c

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

The position vectors of the points

A, B, C

and

D

are

3 \hat{i} - 2 \hat{j} - \hat{k}, 2 \hat{i} - 3 \hat{j} + 2 \hat{k}, 5 \hat{i} - \hat{j} + 2 \hat{k}

and

4 \hat{i} - \hat{j} + λ \hat{k},

respectively. If the points

A, B, C

and

D

lie on a plane, the value of

λ

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

\vec{a} = \hat{j} - \hat{k}

and

\vec{c} = \hat{i} - \hat{j} - \hat{k}

. Then the vector

\vec{b}

satisfying

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

and

\vec{a} \cdot \vec{b} = 3

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

\vec{r}

is a unit vector satisfying

\vec{r} \times \vec{a} = \vec{b}, | \vec{a} | = 2

and

| \vec{b} | = \sqrt{3},

then one such

\vec{r} =

HARD

Mathematics>Vectors and 3D Geometry>Vector Algebra>Linearly Dependent and Independent Vectors

Let

\vec{α} = (λ - 2) \vec{a} + \vec{b}

and

\vec{β} = (4 λ - 2) \vec{a} + 3 \vec{b},

be two given vectors where vectors

\vec{a}

and

\vec{b}

are non-collinear. The value of

λ

for which vectors

\vec{α}

and

\vec{β}

are collinear, is:

Important Questions on Vector Algebra

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