Three vectors P → , Q → and R → are shown in the figure. Let, S be any point on the vector R → . The distance between the points P and S is b R → . The general relation among vectors P → , Q → and S → are

S → = 1 - b P → + b 2 Q →

S → = 1 - b 2 P → + b Q →

Assertion A : If A , B , C , D are four points on a semi-circular arc with a centre at O such that | A B → | = | B C → | = | C D → | . Then, A B → + A C → + A D → = 4 A O → + O B → + O C → Reason R : Polygon law of vector addition yields A B → + B C → + C D → + A D → = 4 A O → In the light of the above statements, choose the most appropriate answer from the options given below.

Both A and R are correct but R is not the correct explanation of A .

A is correct but R is not correct.

A is not correct but R is correct.

Both A and R are correct and R is the correct explanation of A .

The magnitude of vectors OA → , OB → and OC → in the given figure are equal. The direction of OA → + OB → - OC → with x - axis will be:

tan − 1 ⁡ ( 1 − 3 − 2 ) ( 1 + 3 + 2 )

tan − 1 ⁡ ( 3 − 1 + 2 ) ( 1 + 3 − 2 )

tan − 1 ⁡ ( 3 − 1 + 2 ) ( 1 − 3 + 2 )

tan − 1 ⁡ ( 1 + 3 − 2 ) ( 1 − 3 − 2 )

Match List I with List II. List I List II (a) C → - A → - B → = 0 (i) (b) A → - C → - B → = 0 (ii) (c) B → - A → - C → = 0 (iii) (d) A → + B → = - C → (iv) Choose the correct answer from the options given below:

(a) → (iv), (b) → (iii), (c) → (i), (d) → (ii)

(a) → ( iv ) , (b) → ( i ) , (c) → ( iii ) , (d) → (ii)

(a) → (iii), (b) → (ii), (c) → (iv), (d) → (i)

(a) → (i), (b) → (iv), (c) → (ii), (d) → (iii)

Statement I: If three forces F → 1 , F → 2 and F → 3 are represented by three sides of a triangle and F 1 → + F 2 → = - F → 3 , then these three forces are concurrent forces and satisfy the condition for equilibrium. Statement II: A triangle made up of three forces F 1 → , F 2 → and F 3 → as its sides were taken in the same order, satisfies the condition for translatory equilibrium. In the light of the above statements, choose the most appropriate answer from the options given below:

Both Statement I and Statement II are true.

Statement I is true but Statement II is false.

Both Statement I and Statement II are false.

Statement I is false but Statement II is true.

Statement-I : Two forces P → + Q → and P → - Q → where P → ⊥ Q → , when act at an angle θ 1 each other, the magnitude of their resultant is 3 P 2 + Q 2 , when they act at an angle θ 2 , the magnitude of their resultant becomes 2 P 2 + Q 2 . This is possible only when θ 1 < θ 2 . Statement-II : In the situation given above. θ 1 = 60 ° and θ 2 = 90 ° In the light of the above statement, choose the most appropriate answer from the options given below :

Both Statement I and Statement II are true.

Statement I is false but Statement II is true.

Both Statement I and Statement II are false.

Statement I is true but Statement II is false.

EASY

Earn 100

If two forces act in opposite direction then their resultant is $10 N$ and if they act mutually perpendicular then their resultant is $50 N$ . Find the magnitudes of both the forces.

Important Questions on Motion in a Plane

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

A vector

\vec{A}

is rotated by a small angle

∆ θ

radians

(∆ θ ≪ 1)

to get a new vector

\vec{B}

. In that case

|\vec{B} - \vec{A}|

is :

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Two forces

P

and

Q

, of magnitude

2 F

and

(1)

, respectively, are at an angle

θ

with each other. If the force

Q

is doubled, then their resultant also gets doubled. Then, the angle

θ

is:

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Two vectors

\vec{P}

and

\vec{Q}

have equal magnitudes. If the magnitude of

\vec{P} + \vec{Q}

n

times the magnitude of

\vec{P} - \vec{Q},

then angle between

\vec{P}

and

\vec{Q}

HARD

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Three vectors

\vec{P}, \vec{Q}

and

\vec{R}

are shown in the figure. Let,

S

be any point on the vector

\vec{R}

. The distance between the points

P

and

S

b |\vec{R}|

. The general relation among vectors

\vec{P}, \vec{Q}

and

\vec{S}

are

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

\vec{A}

and

\vec{B}

are two non-zero vectors inclined at an angle

θ

\hat{a}

and

\hat{b}

are unit vectors along

\vec{A}

and

\vec{B}

respectively. The component of

\vec{A}

in the direction of

\vec{B}

HARD

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

$Assertion A :$ If $A, B, C, D$ are four points on a semi-circular arc with a centre at $O$ such that $| \vec{A B} | = | \vec{B C} | = | \vec{C D} | .$ Then, $\vec{A B} + \vec{A C} + \vec{A D} = 4 \vec{A O} + \vec{O B} + \vec{O C}$

$Reason R :$ Polygon law of vector addition yields $\vec{A B} + \vec{B C} + \vec{C D} + \vec{A D} = 4 \vec{A O}$

Question Image

In the light of the above statements, choose the most appropriate answer from the options given below.

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The angle between two vectors

x

and

y

θ

. If the resultant vector

z

makes an angle

\frac{θ}{2}

with

x

, then which of the following is true?

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The magnitude of vectors $\vec{OA}, \vec{OB}$ and $\vec{OC}$ in the given figure are equal. The direction of $\vec{OA} + \vec{OB} - \vec{OC}$ with $x -$ axis will be:

Question Image

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Figure shows three forces ${\vec{F}}_{1}, {\vec{F}}_{2}$ and ${\vec{F}}_{3}$ acting along the sides of an equilateral triangle. If the total torque acting at point $O$ (centre of the triangle) is zero then the magnitude of ${\vec{F}}_{3}$ is

Question Image

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

\vec{A} = 3 \hat{i} - 2 \hat{j} + \hat{k}, \vec{B} = \hat{i} - 3 \hat{ȷ} + 5 \hat{k}

and

\vec{C} = 2 \hat{i} + \hat{j} - 4 \hat{k}

form a right angled triangle then out of the following which one is satisfied?

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The angle between vector $(\vec{A})$ and $(\vec{A} - \vec{B})$ is :

Question Image

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Vector

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

and

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

. What is the unit vector along

\vec{a} + \vec{b}

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The resultant of these forces $\vec{O P}, \vec{O Q}, \vec{O R}, \vec{O S}$ and $\vec{O T}$ is approximately ______ $N .$
[Take $\sqrt{3} = 1.7, \sqrt{2} = 1.4$ Given $\hat{i}$ and $\hat{j}$ unit vectors along $x, y$ axis]

Question Image

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Match List I with List II.

List I		List II
(a)	$\vec{C} - \vec{A} - \vec{B} = 0$	(i)
(b)	$\vec{A} - \vec{C} - \vec{B} = 0$	(ii)
(c)	$\vec{B} - \vec{A} - \vec{C} = 0$	(iii)
(d)	$\vec{A} + \vec{B} = - \vec{C}$	(iv)

Choose the correct answer from the options given below:

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Statement I: If three forces

{\vec{F}}_{1}, {\vec{F}}_{2}

and

{\vec{F}}_{3}

are represented by three sides of a triangle and

\vec{F_{1}} + \vec{F_{2}} = - {\vec{F}}_{3},

then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement II: A triangle made up of three forces

\vec{F_{1}}, \vec{F_{2}}

and

\vec{F_{3}}

as its sides were taken in the same order, satisfies the condition for translatory equilibrium.
In the light of the above statements, choose the most appropriate answer from the options given below:

EASY

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

The square of resultant of two equal forces is three times their product. Angle between the forces is?

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Statement-I : Two forces

(\vec{P} + \vec{Q})

and

(\vec{P} - \vec{Q})

where

\vec{P} ⊥ \vec{Q},

when act at an angle

θ_{1}

each other, the magnitude of their resultant is

\sqrt{3 (P^{2} + Q^{2})},

when they act at an angle

θ_{2},

the magnitude of their resultant becomes

\sqrt{2 (P^{2} + Q^{2})} .

This is possible only when

θ_{1} < θ_{2} .

Statement-II : In the situation given above.

θ_{1} = 60 °

and

θ_{2} = 90 °

In the light of the above statement, choose the most appropriate answer from the options given below :

HARD

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

In an octagon $A B C D E F G H$ of equal side, what is the sum of $\vec{A B} + \vec{A C} + \vec{A D} + \vec{A E} + \vec{A F} + \vec{A G} + \vec{A H},$ if, $\vec{AO} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$

Question Image

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Resultant of two vectors

\vec{P}

and

\vec{Q}

is of magnitude

R_{1} .

If direction of

\vec{Q}

is reversed, the resultant is of magnitude

R_{2}

. The value of

(R_{1}^{2} + R_{2}^{2})

[\cos (π - θ) = - \cos θ]

MEDIUM

Physics>Mechanics>Motion in a Plane>Scalars and Vectors

Two vectors

\vec{A}

and

\vec{B}

have equal magnitudes. The magnitude of

(\vec{A} + \vec{B})

n

times the magnitude of

(\vec{A} - \vec{B})

. The angle between

\vec{A}

and

\vec{B}

is:

If two forces act in opposite direction then their resultant is 10 N and if they act mutually perpendicular then their resultant is 50 N. Find the magnitudes of both the forces.

Important Questions on Motion in a Plane

Learn and Prepare for any exam you want !

If two forces act in opposite direction then their resultant is $10 N$ and if they act mutually perpendicular then their resultant is $50 N$ . Find the magnitudes of both the forces.