Scalars and vectors

(A) concept of scalars and as a physical quantities with magnitude and no direction

A scalar quantity is one which has only magnitude, for example, distance, speed, temperature and volume. Scalars can be added and subtracted easily by using ordinary algebra. Vectors, however, have to be added and subtracted by other methods, using, for example, vector diagrams.

(B) concept of vectors as physical quantities with both magnitude and direction

A vector quantity is one which has direction as well as a magnitude. For example, displacement ( which is defined as distance in a given direction), velocity (which is defined as speed in a given direction) and force (which is specified by stating its magnitude and the direction of its line of action). The weight of a body, for example, is a force which acts in a vertical direction downwards.

(C) representation of a vector

          representing a vector

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(D) addition of two vectors (compounding two vectors)

In general, two or more vectors acting on a body in given directions can be combined to give a single vector which produces the same effect. The single vector is called the resultant. Let us consider two forces of magnitude P = 30N and Q = 40N, acting on a body O in any of the following possible directions: 

(I) P and Q act in the same direction;

(II) P and Q act in opposite directions;

(III) P and Q act at right angles to each other;

(IV) P and Q act at an acute angle O = 60°

(V) P and Q act at an obtuse angle O = 120°.

Note: the parallelogram law of vectors states that if two vectors are represented in magnitude and direction by the adjacent sides of a parallelogram, the resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the common point.

(E) resolution of a vector

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(F) the resultant of more than two vectors

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Note: the magnitude of the resultant (from Pythagoras theorem) is given by 

R = √p² + q² 

For the direction of the resultant of a  given line. It is given by:

        Tan O = opposite side/ adjacent side

Past questions

1. Which of the following physical quantities is NOT a vector? (jamb 1980) A. Velocity B. Work C. Force D. Electric field intensity E. Magnetic induction

Answer: B

Solution: vector has magnitude and direction. But work only has magnitude and no direction

2. Which of the following is NOT a vector quantity? (Jamb 1983)

A. Force

B. Altitude

C. Weight

D. Displacement

E. Acceleration

Answer: B

Solution: altitude consists of magnitude with no direction.

3.which of the following quantities are scalars? (Jamb 1988)

I. Electrical potential

II. Torque

III. Momentum

IV. Kinetic energy

A. II and III only

B. I and II only

C. III and IV only

D. I and IV only

Answer: D

Solution: scalar quantity has to do with magnitude only, which makes electrical potential and kinetic scalar quantity.

4.

In the diagram above, PQ and R are vectors. Which of the following options gives the correct relationship between the vectors? (Jamb 1995)

A. P = Q + R

B. P = Q – R

C. P = R – Q

D. P + Q + R = O 

Answer: D

Solution: the correct relationship between them is p+q+r=o ;cos they all act in opposite directions.

5. Which of the following pairs of physical quantities comprises vectors?(wassce 1998)

A. Capacitance and inductance

B. Force ratio and velocity ratio

C. Friction and momentum

D. Energy and power 

E. Electric field potential and electric field intensity

Answer: E

Solution: electric field potential and electric field intensity consists of magnitude and direction    

6. The resultant of two forces acting on an object is maximum when the angle between them is (wassce 2000)

A. 180°

B. 90°

C. 45°

D. 0°

Answer: D

7. Each of the following physical quantities is classified as a vector or a scalar quantity. Which of the classifications is  correct? (Wassce 2005)

A. Electric potential (vector)

B. Momentum (scalar)

C. Gravitational field intensity (scalar)

D. Magnetic flux density (vector)

Answer: B

8. Which of the following pairs of physical quantities is made up of vectors? (Wassce 2014)

A. Speed and displacement

B. Mass and force

C. Displacement and acceleration

D. Momentum and length

Answer: C

Solution: displacement and acceleration has magnitude and direction