Test: Vector - Grade 12 MCQ

Concept:
For two vectors and the resultant will be:

Calculation:
Given:

Concept:
If A and B represents two forces acting at a point making an angle θ, then their resultant is given by:

• Rmax = A + B
• Rmin = A - B

Calculation:
Let A and B be the required forces.
According to the question, maximum and minimum value of the resultant of two forces are 40 units and 20 units respectively.
∴ R_max = A + B = 40...(i) and
R_min = A - B = 20...(ii)
Adding (i) and (ii), we get:
2A = 60
⇒ A = 30 units
⇒ B = 40 - A = 40 - 30 = 10 units
∴ The required forces are 30, 10 units.
The correct answer is Option A. 

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

• Time is a scalar quantity.
• Volume is a scalar quantity.
• Speed is a scalar quantity.
• 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.

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

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

Now squaring both sides
2500 = 900 + F²

So option c is correct.

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.

Vector, V = -3i + 5j
Magnitude of the vector, V;
|V| = √((-3)² + 5²) = √(9+25) = √34

Given, v = 1/√3 i + 1/√3 j + 1/√3 k

Magnitude of v = 1/√3 i + 1/√3 j + 1/√3 k;

|v| = √((1/√3)² + (1/√3)² + (1/√3)²) = √(1+1+1) = √3

Let A = 5i + j – 3k

B = 3i – 4j + 7k

A . B = (5i + j – 3k).(3i – 4j + 7k)

= 5 · 3 + 1 · (-4) + (-3) · 7

= 15 – 4 – 21

= -10