If e → 1 and e → 2 are two unit vectors such that e → 1 - e → 2 is also a unit vector, then find the angle θ between e → 1 and e → 2 .

To find: Magnitude of e 1 → - e 2 → . | e 1 → - e 2 → | = e 1 → 2 + e 2 → 2 - 2 | e 1 → | | e 2 → | cos θ . . . 1 θ is the angle between e 1 → & e 2 → . e 1 → = e 2 → = 1 (from question) | e 1 → - e 2 → | = 1 (from question) From equation 1 , 1 = 1 + 1 - 2 cos θ ⇒ - 1 = - 2 cos θ ⇒ cos θ = 1 2 ⇒ θ = 60 °

If e → 1 and e → 2 are two unit vectors such that e → 1 - e → 2 is also a unit vector, then find the angle θ between e → 1 and e → 2 .

To find: Magnitude of e 1 → - e 2 → . | e 1 → - e 2 → | = e 1 → 2 + e 2 → 2 - 2 | e 1 → | | e 2 → | cos θ . . . 1 θ is the angle between e 1 → & e 2 → . e 1 → = e 2 → = 1 (from question) | e 1 → - e 2 → | = 1 (from question) From equation 1 , 1 = 1 + 1 - 2 cos θ ⇒ - 1 = - 2 cos θ ⇒ cos θ = 1 2 ⇒ θ = 60 °

Find a vector in the direction of the vector a → = i ^ - 2 j ^ that has magnitude of 7 units.

To find a vector in the direction of vector a → with a magnitude of 7 units, we first need to find the unit vector in the direction of vector a → . The unit vector is given as, u a → = a → a → ⇒ u a → = i ^ - 2 j ^ 1 2 + - 2 2 ⇒ u a → = i ^ - 2 j ^ 1 + 4 ⇒ u a → = i ^ - 2 j ^ 5 ⇒ u a → = 1 5 i ^ - 2 5 j ^ Now, we have got a unit vector in the direction of a → . The vector of a magnitude of 7 units in the direction of a → can be obtained as, 7 u a → = 7 1 5 i ^ - 2 5 j ^ ⇒ 7 u a → = 7 5 i ^ - 14 5 j ^

HARD

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If ${\vec{e}}_{1}$ and ${\vec{e}}_{2}$ are two unit vectors such that ${\vec{e}}_{1} - {\vec{e}}_{2}$ is also a unit vector, then find the angle $θ$ between ${\vec{e}}_{1}$ and ${\vec{e}}_{2} .$

Important Questions on Vector Algebra

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

Number of unit vectors of the form

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

where

a, b, c \in W

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

What is the vector

r

of magnitude

2 \sqrt{3}

units that makes an angle of

\frac{π}{2}

and $\frac{\pi}{6}$ with

y

-axis and

z

-axis respectively?

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

The direction cosines of the vector

\hat{i} - 5 \hat{j} + 8 \hat{k}

are

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

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

The vector that is parallel to the vector

2 \hat{i} - 2 \hat{j} - 4 \hat{k}

and coplanar with the vectors

\hat{i} + \hat{j}

and

\hat{j} + \hat{k}

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

A unit vector is represented as

(0 .8 \hat{i} + b \hat{j} + 0 .4 \hat{k})

. Hence the value of

b

must be

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

The values of

α

such that

| α \hat{i} + (α + 1) \hat{j} + 2 \hat{k} | = 3

, are

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

Let

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

\vec{b} = \vec{c} - \vec{d}, \vec{a}

is parallel to

\vec{c}

and perpendicular to

\vec{d}

, then

\vec{c} + \vec{d} =

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

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is

\sqrt{3}

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

Let a vector

α \hat{i} + β \hat{j}

be obtained by rotating the vector

\sqrt{3} \hat{i} + \hat{j}

by an angle

45 °

about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices

(α, β), (0, β)

and

(0, 0)

is equal to

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

Find a vector in the direction of the vector

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

that has magnitude of

7

units.

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 :

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

MEDIUM

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

P Q R S T

is a pentagon, then the resultant of forces

\vec{P Q}, \vec{P T}, \vec{Q R}, \vec{S R}, \vec{T S}

and

\vec{P S}

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

If the vectors

x \hat{i} - 3 \hat{j} + 7 \hat{k}

and

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

are collinear then the value of

\frac{x y^{2}}{z}

is equal

EASY

Mathematics>Coordinate Geometry>Vector Algebra>Basics of Vectors

\vec{a}

is a nonzero vector of magnitude

‘ a ’

and

λ

a nonzero scalar then

λ \vec{a}

is unit vector if

If e→1 and e→2 are two unit vectors such that e→1-e→2 is also a unit vector, then find the angle θ between e→1 and e→2.

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

If ${\vec{e}}_{1}$ and ${\vec{e}}_{2}$ are two unit vectors such that ${\vec{e}}_{1} - {\vec{e}}_{2}$ is also a unit vector, then find the angle $θ$ between ${\vec{e}}_{1}$ and ${\vec{e}}_{2} .$