Vectors MCQ Quiz - Objective Question with Answer for Vectors - Download Free PDF

option(3)

CONCEPT:

• Vector quantities are those physical quantities that have both magnitude and direction and obey the vector law of addition are called vector quantities or vectors.
  • Example Displacement, Velocity, Force, Momentum, etc
• If a vector is multiplied by a real number then both the magnitude and direction of the vector remain change.

EXPLANATION:

• Multiplication of a vector by a real number
• If a vector \(\vec A\) is multiplied by the real number λ = 2 then we get another vector \(\vec B\) such that \(\vec B\) = 2\(\vec A\), as shown in the figure you can see that the magnitude of \(\vec B\) is twice the magnitude of \(\vec A\) while the direction of \(\vec B\) is same as that of \(\vec A\)

Additional Information

• If a vector \(\vec A\)is multiplied by real number λ = -2, then the new vector is such that \(\vec B\)= -2 \(\vec A\). As shown in the above figure the magnitude of \(\vec B\)is again twice the magnitude of \(\vec A\)but the direction is opposite to that of \(\vec A\)
• Hence we understood that If a vector is multiplied by a real number then both the magnitude as well as the direction of the vector remain change.

Given:

\(\vec{a}=\hat{i}+\hat{j}+\hat{k}\)

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

\(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\)

Calculation:

The unit vector in the direction of the vector,  \(\vec r = 2\vec{a}-\vec{b}+3\vec{c}\)

⇒ \(\vec r = 2(\hat i + \hat j + \hat k) - (2\hat i - \hat j + 3\hat k) + 3(\hat i - 2\hat j + \hat k)\)

⇒ \(\vec r = 2 \hat i + 2\hat j + 2\hat k - 2 \hat i + \hat j - 3\hat k + 3 \hat i - 6 \hat j + 3 \hat k\)

⇒ \(\vec r = (2 - 2 + 3) \hat i + (2 + 1 - 6) \hat j + (2 - 3 + 3) \hat k\)

⇒ \(\vec r = 3 \hat i - 3 \hat j + 2 \hat k\)

Magnitude or \(\vec r\) = \(\sqrt{3^2 + (-3)^2 + 2^2}\)

⇒ \(|\vec r|\) = \(\sqrt{9+ 9+ 4}\) = \(\sqrt{22}\)

Unit vector in the direction of \(\vec r = \frac{1}{| \vec r|}\times \vec r\)

⇒ \(\vec r = \frac{1}{\sqrt{22}} \times [3 \hat i - 3 \hat j + 2 \hat k]\)

⇒ \(\vec r = \frac{3}{\sqrt{22}} \hat i - \frac{3}{\sqrt{22}} \hat j + \frac{2}{\sqrt{22}} \hat k\)

∴ The required vector is \(\frac{3}{\sqrt{22}} \hat i - \frac{3}{\sqrt{22}} \hat j + \frac{2}{\sqrt{22}} \hat k\)

Concept:

Angle Between Two Equal Vectors:

• When two vectors are equal in magnitude, the angle between them is zero, as they point in the same direction.
• The angle between two equal vectors can be derived from the properties of vector addition, where the magnitude of the resultant vector is the sum of the magnitudes of the two vectors.
• If two vectors are in the same direction, the angle between them is 0°. This is because the cosine of 0° is 1, resulting in a maximum dot product.

Calculation:

For two equal vectors A and B, the angle between them is:

cos(θ) = (A · B) / (|A| |B|)

Since the vectors are equal and pointing in the same direction, θ = 0°.

Conclusion:

The angle between the two equal vectors is 0°.

Calculation: 

\(\vec{\tau}=\vec{r} \times \vec{F}\)

= \((2 \hat{i}+\hat{j}+2 \hat{k}) \times(3 \hat{i}+4 \hat{j}-2 \hat{k})\)

= \(-10 \hat{i}+10 \hat{j}+5 \hat{k}\)

∴ the torque about the origin is \(-10 \hat{i}+10 \hat{j}+5 \hat{k}\)

CONCEPT:

• All measurable quantities are divided into two broad categories:

EXPLANATION:

• Energy is quantity has only magnitude. It does not require any direction. So it is a scaler, not vector quantity.
• The weight of a body denotes gravitational force. And force requires magnitude and direction (in which direction it is acting) both to describe.
  • So weight is a vector.
• Similarily momentum requires magnitude and direction (in which direction it is acting) both to describe.
  • So momentum is a vector.
• In chemistry, you will study that the Nuclear spin of an electron has two directions clockwise and anti-clockwise. So it will also be a vector. 

CONCEPT:

• Speed: The rate of change of distance is called speed.
  • It is a scalar quantity.
• Velocity: The rate of change of displacement is called velocity. 
  • It is a vector quantity.
• Scalar quantities: The physical quantities which have only magnitude and no direction are called scalar quantities or scalars.
  • Examples: Mass, volume, density, time, temperature, electric current, Luminious intensity, speed, etc.
• Vector quantities: The physical quantities which have both magnitude and direction and obey the laws of vector addition are called vector quantities or vectors.
  • Examples Displacement, velocity, acceleration, force, momentum, Impulse, etc.

EXPLANATION:

1. Time is a scalar quantity.
2. Volume is a scalar quantity.
3. Speed is a scalar quantity.
4. Velocity is a vector quantity. So option 4 is correct.

CONCEPT:

• All measurable quantities are divided into two broad categories:

EXPLANATION:

• From the above table, it is clear that force, velocity, and acceleration are vector quantity because they have both magnitude as well as direction. Therefore option 1, 2, and 3 is incorrect.
• Pressure is defined as force per unit area. It is a scalar quantity because it has only magnitude and it is independent on the size of the area chosen. Therefore option 4 is correct.

The correct answer is 40N

CONCEPT:​

• Resolution of vectors into components: We have a vector (F) where the magnitude of the vector is F and the angle with horizontal is θ.

The vector has two components: 1. Vertical component and 2. Horizontal component

Vertical component (F_y) = F Sinθ

Horizontal component (F_x) = F Cosθ 

Here \(F = \sqrt {F_x^2 + F_y^2}\)

CALCULATION:

Here F₁ and F₂ are along X- and Y- direction.

Let the applied force F = 50

And the x-component of the applied force F_x = 30

The y-component of the applied force F_y = ?

We know that the vector sum of the force

\(F = \sqrt {F_x^2 + F_y^2}\)

\(50N = \sqrt {{{30}^2} + {F^2}}\)

Now squaring both sides

2500 = 900 + F²

\({F_y} = \sqrt {2500 - 900} = \sqrt {1600}\)

\({F_y} = 40N\)

So option 3 is correct.

CONCEPT:

• Speed: The rate of change in distance is called speed. 
  • It is a scalar quantity. It is the magnitude of velocity that can never be negative.
• Mass: The quantity of matter in any object is called the mass of that object.
  • It can never be negative. It is a scalar quantity.
• Velocity: The rate of change in displacement is called velocity.
  • It is a vector quantity that can be negative, positive, or zero.
• Distance: The total path length between two points is called distance.
  • It is a scalar quantity and can never be negative.

EXPLANATION:

• As discussed above, velocity is a vector quantity that can be negative. So option 3 is correct.

Additional Information

• Scalar quantities: The physical quantities which have only magnitude and no direction are called scalar quantities or scalars.
  • A scalar quantity can be specified by a single number, along with the proper unit.
  • Examples: Mass, volume, density, time, temperature, electric current, Luminious intensity, etc.
• Vector quantities: The physical quantities which have both magnitude and direction and obey the laws of vector addition are called vector quantities or vectors.
  • A vector quantity is specified by a number with a unit and its direction.
  • Examples Displacement, velocity, force, momentum, etc

CONCEPT:

• All measurable quantities are divided into two broad categories:

EXPLANATION:

• From the above, it is clear that mass, length, and speed are scalar quantity because they have the only magnitude. Therefore option 1, 2, and 3 is incorrect.

• Impulse is a vector quantity because the force is a vector quantity. So option 4 is an example of a vector quantity.

CONCEPT:

• Acceleration (a): The rate of change of the velocity of an object is called acceleration.
  • It is a vector quantity.
• Scalar quantities: The physical quantities which have only magnitude and no direction are called scalar quantities or scalars.
  • Examples: Mass, volume, density, time, temperature, electric current, Luminious intensity, etc.
• Vector quantities: The physical quantities which have both magnitude and direction and obey the laws of vector addition are called vector quantities or vectors.
  • Examples Displacement, velocity, acceleration, force, momentum, Impulse, etc.

EXPLANATION:

• Acceleration is a vector quantity. So option 1 is correct.

CONCEPT:

• Force: The interaction which after applying on a body changes or try to change the state of rest or state of motion is called force.
  • It is a vector quantity.
• Temperature: The measurement of hotness is called temperature.
  • It is a scalar quantity.
• Scalar quantities: The physical quantities which have only magnitude and no direction are called scalar quantities or scalars.
  • Examples: Mass, volume, density, time, temperature, electric current, Luminious intensity, etc.
• Vector quantities: The physical quantities which have both magnitude and direction and obey the laws of vector addition are called vector quantities or vectors.
  • Examples Displacement, velocity, acceleration, force, momentum, Impulse, etc.

EXPLANATION:

1. Displacement is a vector quantity as it is directed from the initial point to the final point. 
2. Current: The rate of flow of electric charge is called current.
3. Temperature is NOT a vector quantity. It is a scalar quantity. So option 2 is correct.
4. Drag is also a type of force and it is a vector quantity.
5. Force is a vector quantity.

CONCEPT:

The dot product of vector:

• The dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
• Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.

\({{\rm{A}}_1}\cdot{{\rm{A}}_2} = \left| {\overrightarrow {{{\rm{A}}_1}} } \right|\left| {\overrightarrow {{{\rm{A}}_2}} } \right|\cos {\rm{θ }}\)

Where \(\left| {\overrightarrow {{{\rm{A}}_1}} } \right|.\left| {\overrightarrow {{{\rm{A}}_2}} } \right|\) are the magnitudes of two vectors A1 and A2

EXPLANATION:

• The angle between two vectors \(\vec A\) and \(\vec B\) given by

\(\Rightarrow cos\;\theta = \frac{{\vec A.\vec B}}{{\left| {\vec A} \right|\left| {\vec B} \right|}}\)

• Therefore option 4 is correct.

Concept:

Physical quantities are of two types:

Explanation:

Electric Field Intensity (E):

• The space around an electric charge in which its influence can be felt is known as the electric field.
• The electric field intensity at a point is the force experienced by a unit positive charge placed at that point.
• Electric Field Intensity is a vector quantity. It is denoted by ‘E’.
• Electric Field = F/q.
• Unit of E is NC-1 or Vm-1

Electrostatic energy and electrostatic potential are scalar quantities because it only requires magnitude and not the direction.

Mistake Points

• In the case of electric current, when two currents meet at a junction, the resultant current of these will be an algebraic sum and not the vector sum.
• Therefore, an electric current is a scalar quantity although it possesses magnitude and direction.