Two vectors a → and b → , inclined at an angle θ with respect to each other, have a resultant c → , which makes an angle β with a → . If the directions of a → and b → are interchanged, then the resultant will have the same

magnitude as well as direction.

neither magnitude nor direction.

Can A → + B → = A → - B → ?

Yes, it is possible when both the vectors are perpendicular to each other. The magnitude of addition and subtraction of two vectors will be the same under this condition because even if we reverse the direction of a vector they will still be perpendicular to each other. Note: We know that : 1. The subtraction of a vector B → from vector A → can be treated as reversing the direction of B → and then adding it to A → . 2. Resultant of two vectors is expressed as: R → = A 2 + B 2 + 2 A B cos θ Hence, when both the vectors are perpendicular then R → will remain same in magnitude for addition or subtraction of the two given vectors.

Establish the following vector inequalities geometrically or otherwise: (b) | a + b | ≥ | | a | - | b | | When does the equality sign above apply?

The two vectors a → and b → can be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure. O M → = a → M N → = O P → = b → O N → = a → + b → In a triangle, the sum of the length of the two sides is greater than the length of the 3 rd side. Therefore, in ΔOMN, we have: O N + M N > O M O N > O M - M N ∵ O P = M N ∴ O N → > O M → - O P → a → + b → > a → - b → … (i) When two vectors a → and b → act along a straight line in the opposite direction, then we can write: a → + b → = a → - b → ….. (ii) From equation (i) & (ii) we can write: a → + b → ≥ a → - b →

EASY

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Can the resultant of 2 vectors be zero ?

Important Questions on Scalars and Vectors

HARD

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

In an octagon $r'$ 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

EASY

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of 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?

EASY

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of 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

MEDIUM

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

Two vectors

\vec{a}

and

\vec{b}

, inclined at an angle

θ

with respect to each other, have a resultant

\vec{c}

, which makes an angle

β

with

\vec{a}

. If the directions of

\vec{a}

and

\vec{b}

are interchanged, then the resultant will have the same

MEDIUM

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

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

, then the angle between

\vec{A}

and

\vec{B}

MEDIUM

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

Can

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

MEDIUM

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

Establish the following vector inequalities geometrically or otherwise:

(b) $| a + b | \geq | | a | - | b | |$

When does the equality sign above apply?

MEDIUM

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

The figure shows three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ . Here, $R$ is the midpoint of $P Q$ . Then which of the following relations is correct?

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EASY

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

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

MEDIUM

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

$A B C D E F$ is a regular hexagon. What is the value of $\vec{A B} + \vec{A C} + \vec{A D} + \vec{A E} + \vec{A F}$ ?

Question Image

EASY

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

The sum and difference of two perpendicular vectors, of equal lengths, are

EASY

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

A car is moving on a straight road, due north, with a uniform speed of

50 km h^{- 1}

, when it turns left through

90 °

. If the speed remains unchanged after turning, the change in the velocity of the car, in the turning process, is

HARD

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

Define parallelogram law of vector addition. Calculate the magnitude and direction of the resultant vector of two vectors

P

and

Q

acting at angle

θ

. What happen to the resultant vector if the given vectors are collinear or anti-collinear in nature?

EASY

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

Given,

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

and

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

. The unit vector of

\vec{A} - \vec{B}

EASY

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

\vec{A}

and

\vec{B}

denote the sides of a parallelogram and its area is

\frac{1}{2} A B

(

A

and

B

are magnitude of

\vec{A}

and

\vec{B}

respectively), the angle between

\vec{A}

and

\vec{B}

MEDIUM

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

A bus is moving on a straight road towards north with a uniform speed of

50 km h^{- 1}

turns through

90 °

anticlockwise. If the speed remains unchanged after turning , the increase in the velocity of bus in the turning process is:

EASY

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

The resultant of two vectors of magnitude

2 A

and

\sqrt{2} A

, acting at an angle

θ

, is

\sqrt{10} A

. Find the value of

θ

EASY

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

If the angle between two force increases, the magnitude of their resultant

MEDIUM

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

Given,

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

. Also, the magnitude of

\vec{A}

\vec{B}

and

\vec{C}

are

12

5

and

13 units

, respectively. The angle between

\vec{A}

and

\vec{B}

MEDIUM

Physics>Mechanics>Scalars and Vectors>Addition and Subtraction of Vectors

Let the vectors

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

and

\vec{U P}

represent the sides of a regular hexagon.

Statement I:

\vec{P Q} \times (\vec{R S} + \vec{S T}) \neq \vec{0}

Statement II:

\vec{P Q} \times \vec{R S} = \vec{0}

and

\vec{P Q} \times \vec{S T} \neq \vec{0}

Can the resultant of 2 vectors be zero ?

Important Questions on Scalars and Vectors

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