EASY

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Is it possible that dot product of two vectors is zero even if they are not perpendicular to each other ?

Important Questions on Work, Energy and Power

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

If vectors

\vec{A} = \cos ω t \hat{i} + s i n ω t \hat{j}

and

\vec{B} = c o s \frac{ω t}{2} \hat{i} + \sin \frac{ω t}{2} \hat{j}

are functions of time, then the value of

t

at which they are orthogonal to each other is:

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

Two vectors

\vec{X}

and

\vec{Y}

have equal magnitude. The magnitude of

(\vec{X} - \vec{Y})

n

times the magnitude of

(\vec{X} + \vec{Y})

. The angle between

\vec{X}

and

\vec{Y}

is :

MEDIUM

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

Three particles

P, Q

and

R

are moving along the vectors

\vec{A} = \hat{i} + \hat{j}, \vec{B} = \hat{j} + \hat{k}

and

\vec{C} = - \hat{i} + \hat{j}

, respectively. They strike on a point and start to move in different directions. Now particle

P

is moving normal to the plane which contains vector

\vec{A}

and

\vec{B} .

Similarly particle

Q

is moving normal to the plane which contains vector

\vec{A}

and

\vec{C} .

The angle between the direction of motion of

P

and

Q

\cos^{- 1} (\frac{1}{\sqrt{x}}) .

Then the value of

x

.

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

Let

\vec{A} = (\hat{i} + \hat{j}) a n d \vec{B} = (2 \hat{i} - \hat{j})

. The magnitude of a coplanar vector

\vec{C}

such that

\vec{A} . \vec{C} = \vec{B} . \vec{C} = \vec{A} . \vec{B}

is given by:

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

The vector sum of two forces is perpendicular to their vector differences. In that case, the forces

MEDIUM

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

The vectors

2 r \hat{i} + 3 r \hat{j} + 4 r \hat{k}, a r \hat{i} + b r \hat{j} + c r \hat{k}

are normal to each other only if

MEDIUM

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

What will be the projection of vector

\vec{A} = \hat{i} + \hat{j} + \hat{k}

on vector

\vec{B} = \hat{i} + \hat{j} ?

MEDIUM

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

\vec{A} = (2 \hat{i} + 3 \hat{j} - \hat{k}) m

and

\vec{B} = (\hat{i} + 2 \hat{j} + 2 \hat{k}) m

. The magnitude of component of vector

\vec{A}

along vector

\vec{B}

will be _____

m

HARD

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

In a triangle

A B C

, the sides

A B

and

A C

are represented by the vectors

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

and

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

respectively. Calculate the angle

∠ A B C .

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

\vec{P} = 2 \hat{ı} - 2 \hat{j} + 3 \hat{k}

and

\vec{Q} = \hat{ı} + \hat{j} + 2 \hat{k}

, then cosine of angle between

\vec{P}

and

\vec{Q}

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

Two vectors have magnitude

3

and

5

. If the angle between them is

60 °

, then the dot product of two vectors will be

MEDIUM

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

The sum of two forces

\vec{P}

and

\vec{Q}

\vec{R}

such that

|\vec{R}| = |\vec{P}|

. The angle

θ

(in degree) that the resultant of

2 \vec{P}

and

\vec{Q}

will make with

\vec{Q}

is, ________

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

Which of the following relations is true for two unit vector

\hat{A}

and

\hat{B}

making an angle

θ

to each other?

MEDIUM

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

Find the angle between the two vectors:

\vec{a} = 3 \hat{i} + 2 \hat{j} + 5 \hat{k}

\vec{b} = 5 \hat{i} + 3 \hat{j} + \hat{k}

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

Two vectors

\vec{A}

and

\vec{B}

have equal magnitudes. If magnitude of

\vec{A} + \vec{B}

is equal to two times the magnitude of

\vec{A} - \vec{B}

, then the angle between

\vec{A}

and

\vec{B}

will be

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

An ant starts from the origin and crawls

10 cm

along the

x

-axis and then

20 cm

along the

y

-axis. The dot product of the ant's displacement vector with the position vector of a point that makes

45 °

with the

x

-axis and has a magnitude of

\sqrt{2} cm

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

If the projection of

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

\hat{i} + 2 \hat{j} + α \hat{k}

is zero. Then, the value of

α

will be

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

A triangular plate is shown. A force $\vec{F} = 4 \hat{i} - 3 \hat{j}$ is applied at point $P .$ The torque at point $P$ with respect to point $O$ and $Q$ are:

Question Image

MEDIUM

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

Let

|\vec{A_{1}}| = 3, |\vec{A_{2}}| = 5

and

|\vec{A_{1}} + \vec{A_{2}}| = 5 .

The value of

(2 \vec{A_{1}} + 3 \vec{A_{2}}) \cdot (3 \vec{A_{1}} - 2 \vec{A_{2}})

is:

EASY

Physics>Mechanics>Work, Energy and Power>The Scalar Product of Vectors

If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is:

Is it possible that dot product of two vectors is zero even if they are not perpendicular to each other ?

Important Questions on Work, Energy and Power

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