Let u ^ = u 1 i ^ + u 2 j ^ + u 3 k ^ be a unit vector in R 3 and w ^ = 1 6 i ^ + j ^ + 2 k ^ . Given that there exists a vector v → in R 3 such that u ^ × v → = 1 and w ^ . u ^ × v → = 1 . Which of the following statement(s) is (are) correct ?

If u ^ lies in the xy - plane then u 1 = u 2

There is exactly one choice for such v →

If u ^ lies in the xz - plane then 2 u 1 = u 3

HARD

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Let $a, b, c$ be distinct non-negative numbers and the vectors $a \hat{i} + a \hat{j} + c \hat{k}, \hat{i} + \hat{k}, c \hat{i} + c \hat{j} + b \hat{k}$ lie in a plane, then the quadratic equation $a x^{2} + 2 c x + b = 0$ has -

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Important Questions on Vector Algebra

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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:

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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:

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

Let

\vec{a} = 2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k}, \vec{b} = 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k}

and

\vec{c} = 3 \hat{i} + 6 \hat{j} + (λ_{3} - 1) \hat{k}

be three vectors such that

\vec{b} = 2 \vec{a}

and

\vec{a}

is perpendicular to

\vec{c} .

Then a possible value of

(λ_{1}, λ_{2}, λ_{3})

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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>Coordinate Geometry>Vector Algebra>Vector Equations

Let

\hat{u} = u_{1} \hat{i} + u_{2} \hat{j} + u_{3} \hat{k}

be a unit vector in

R^{3}

and

\hat{w} = \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2 \hat{k}) .

Given that there exists a vector

\vec{v}

R^{3}

such that

|\hat{u} \times \vec{v}| = 1

and

\hat{w} . (\hat{u} \times \vec{v}) = 1 .

Which of the following statement(s) is (are) correct ?

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

Consider the cube in the first octant with sides

O P, O Q

and

O R

of length

1

, along the

x

-axis,

y

-axis and

z

-axis, respectively, where

O (0, 0, 0)

is the origin. Let

S (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})

be the centre of the cube and

T

be the vertex of the cube opposite to the origin

O

such that

S

lies on the diagonal

O T

. If

\vec{p} = \vec{S P}, \vec{q} = \vec{S Q}, \vec{r} = \vec{S R}

and

\vec{t} = \vec{S T}

, then the value of

|(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t})|

is ______.

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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>Coordinate Geometry>Vector Algebra>Vector Equations

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

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

Let

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

\vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + \sqrt{2} \hat{k}

and

\vec{c} = 5 \hat{i} + \hat{j} + \sqrt{2} \hat{k}

be three vectors such that the projection vector of

\vec{b}

\vec{a}

|\vec{a}|

. If

\vec{a} + \vec{b}

is perpendicular to

\vec{c}

, then

|\vec{b}|

is equal to:

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

Let

\vec{a}, \vec{b}

and

\vec{c}

be three unit vectors, out of which vectors

\vec{b}

and

\vec{c}

are non-parallel. If

α

and

β

are the angles which vector

\vec{a}

makes with vectors

\vec{b}

and

\vec{c}

respectively and

\vec{a} \times (\vec{b} \times \vec{c}) = \frac{1}{2} \vec{b}

, then

|α - β|

is equal to :

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

Let $\vec{a}, \vec{b} and \vec{c}$ be three non - zero vectors such that no two of them are collinear and $(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3} |\vec{b}| |\vec{c}| \vec{a}$ . If $θ$ is the angle between vectors $\vec{b} and \vec{c}$ , then a value of $\sin θ$ is

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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

and

\vec{c} = \hat{i} + m \hat{j} + n \hat{k}

are three coplanar vectors and

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

then which one of the following is correct?

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

Let

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

be three vectors having magnitudes

1, 1

and

2

respectively. If

\vec{a} \times (\vec{a} \times \vec{c}) + \vec{b} = \vec{0},

then the angle between

\vec{a}

and

\vec{c}

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

In a parallelogram

A B C D, |\vec{A B}| = a, |\vec{A D}| = b & |\vec{A C}| = c

\vec{D B} \cdot \vec{A B}

has the value:

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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>Coordinate Geometry>Vector Algebra>Vector Equations

\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} =

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Vector Equations

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} =

Let a, b, c be distinct non-negative numbers and the vectors ai^+aj^+ck^, i^+k^, ci^+cj^+bk^ lie in a plane, then the quadratic equation ax2+ 2cx+b=0 has -

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Let $a, b, c$ be distinct non-negative numbers and the vectors $a \hat{i} + a \hat{j} + c \hat{k}, \hat{i} + \hat{k}, c \hat{i} + c \hat{j} + b \hat{k}$ lie in a plane, then the quadratic equation $a x^{2} + 2 c x + b = 0$ has -