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 .

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 →

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.

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.

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 )

EASY

Earn 100

Two forces $F_{1} = 1 N$ and $F_{2} = 2 N$ act along the lines $x = 0$ and $y = 0$ , respectively. Then the resultant of forces would be

50% studentsanswered this correctly

Important Questions on Scalars and Vectors

MEDIUM

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

Consider a frame that is made up of two thin massless rods $A B$ and $A C$ as shown in the figure. A vertical force $\vec{P}$ of magnitude $100 N$ is applied at point $A$ of the frame.

Question Image

Suppose the force is $\vec{P}$ resolved parallel to the arms $A B$ and $A C$ of the frame. The magnitude of the resolved component along the arm $A C$ is $x N$ . The value of $x$ , to the nearest integer, is ________.

[Given : $\sin (35 °) = 0.573, \cos (35 °) = 0.819, \sin (110 °) = 0.939, \cos (110 °) = - 0.342$ ]

MEDIUM

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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 :

HARD

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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

HARD

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

$Assertion A :$ 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.

MEDIUM

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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]

EASY

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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>Scalars and Vectors>Resolution of Vector

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

MEDIUM

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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 :

EASY

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

\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}

EASY

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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>Scalars and Vectors>Resolution of Vector

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>Scalars and Vectors>Resolution of Vector

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:

MEDIUM

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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>Scalars and Vectors>Resolution of Vector

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>Scalars and Vectors>Resolution of Vector

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

MEDIUM

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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

MEDIUM

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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

Question Image

EASY

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

\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?

EASY

Physics>Mechanics>Scalars and Vectors>Resolution of Vector

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>Scalars and Vectors>Resolution of Vector

Two forces

P

and

Q

, of magnitude

2 F

and

3 F

, 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:

Two forces F1=1 N and F2=2 N act along the lines x=0 and y=0, respectively. Then the resultant of forces would be

Important Questions on Scalars and Vectors

Learn and Prepare for any exam you want !

Two forces $F_{1} = 1 N$ and $F_{2} = 2 N$ act along the lines $x = 0$ and $y = 0$ , respectively. Then the resultant of forces would be