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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What is the difference between a fixed vector and a free vector?

Important Questions on Vector Algebra

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of 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>Coordinate Geometry>Vector Algebra>Basics of 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 :

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Basics of 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

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

\vec{a} = \hat{i} + λ \hat{j} + 2 \hat{k} & \vec{b} = μ \hat{i} + \hat{j} - \hat{k}

are orthogonal and

|\vec{a}| = |\vec{b}|

, then

(λ, μ) =

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

Let

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

and

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

be two vectors. Consider a vector

\vec{c} = α \vec{a} + β \vec{b}, α, β \in R .

If the projection of

\vec{c}

on the vector

(\vec{a} + \vec{b})

3 \sqrt{2},

then the minimum value of

(\vec{c} - (\vec{a} \times \vec{b})) . \vec{c}

equals

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

If the vectors

a \hat{i} + \hat{j} + \hat{k}, \hat{i} + b \hat{j} + \hat{k}

and

\hat{i} + \hat{j} + c \hat{k}

are coplanar

(a \neq b \neq c \neq 1)

, then the value of

a b c - (a + b + c) =

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of 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

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

Let

O

be the origin. Let

\vec{O P} = x \hat{i} + y \hat{j} - \hat{k}

and

\vec{O Q} = - \hat{i} + 2 \hat{j} + 3 x \hat{k}, x, y \in R, x > 0,

be such that

| \vec{P Q} | = \sqrt{20}

and the vector

\vec{O P}

is perpendicular to

\vec{O Q} .

\vec{O R} = 3 \hat{i} + z \hat{j} - 7 \hat{k}, z \in R,

is coplanar with

\vec{O P}

and

\vec{O Q},

then the value of

x^{2} + y^{2} + z^{2}

is equal to

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of 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}

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

The position vector of $A$ and $B$ are $2 \hat{i} + 2 \hat{j} + \hat{k}$ and $2 \hat{i} + 4 \hat{j} + 4 \hat{k}$ . The length of the internal bisector of $∠ B O A$ of triangle $A O B$ is

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

Given two vectors

\hat{i} - \hat{j}

and

\hat{i} - 2 \hat{j}

. The unit vector, coplanar with the two given vectors and perpendicular to

(i - j)

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of 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

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

If vectors

\vec{a_{1}} = x \hat{i} - \hat{j} + \hat{k}

and

\vec{a_{2}} = \hat{i} + y \hat{j} + z \hat{k}

are collinear, then a possible unit vector parallel to the vector

x \hat{i} + y \hat{j} + z \hat{k}

is:

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

The unit vector which is orthogonal to the vector

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

and is coplanar with vectors

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

and

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

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

A vector

\vec{a}

has components

3 p

and

1

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

\vec{a}

has components

p + 1

and

\sqrt{10},

then a value of

p

is equal to:

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

If the points with position vectors

30 \hat{i} + 3 \hat{j}, 20 \hat{i} + 6 \hat{j}

and

5 \hat{i} + b \hat{j}

are collinear, then

b

equal

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

If a vector

\vec{x}

makes angles with measure

\frac{π}{4}

and

\frac{5 π}{4}

with positive directions of

X

-axis and

Y

-axis respectively, then

\vec{x}

made angle of measure …... with positive direction of

Z

-axis

HARD

Mathematics>Coordinate Geometry>Vector Algebra>Basics of 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?

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of 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>Coordinate Geometry>Vector Algebra>Basics of 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

What is the difference between a fixed vector and a free vector?

Important Questions on Vector Algebra

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