### Scalar Quantities:

**Scalar quantities** are physical quantities which have magnitude only or numerical value but no direction.

Examples of Scalar Quantities are

- Mass
- Length
- density
- Time
- Distance
- Temperature
- Speed
- Volume
- Density
- Area
- Current

### Addition of Scalar Quantities:

With scalar addition, all you have to do is add the values of each scalar item to obtain the total

Examples of Addition of scalar Quantities:

â€¢ V_{1} + V_{2} = 10 cmÂ³ + 30 cmÂ³ = 40 cmÂ³

â€¢ T_{1} + T_{2} = 273K + 100K = 373K

â€¢ A_{1} + A_{2} = 200 mÂ² + 250 mÂ² = 450 mÂ²

### Vector Quantities:

**Vector quantities** are those quantities that have both magnitude and direction.

Examples of Vector Quantities are

- Force
- Gravitational field
- Momentum
- Weight
- Velocity
- Acceleration due to gravity
- Magnetic field
- Electric field
- Displacement
- Acceleration

### Representation of Vector:

A vector is always represented by a straight line with an arrow at the one end pointing in a particular direction. The length of the line represents the magnitude of the vector e.g.

The vector is 10N in the Eastern direction.

### Addition of Vector Quantities:

**1. **If two vectors \( \scriptsize \vec{B} \) and \( \scriptsize \vec{D} \) moved in the same direction and are 10N and 12N respectively, then, their resultant R is

R = 22N

**2.** If \( \scriptsize \vec{B} \) and \( \scriptsize \vec{D} \) are in the opposite direction.

R = 2N

\( \scriptsize \vec{D} \: -\: \scriptsize \vec{B} = R \)The resultant is in the direction of the larger force i.e. \( \scriptsize \vec{D}\)

### Example 1:

Given that 40 N force F_{1} travels from west to east and 25 N force F_{2} travels in the opposite direction to that of 40 N force. Find the resultant of the two vectors.

**Solution:**

Resultant Vector R = vector F_{1} + ( – vector F_{2})

Resultant vector R = vector F_{1} â€“ vector F_{2}

Resultant vector R = 40 N â€“ 25 N = 15 N

### Example 2:

90 N force travelled from east to west. If 35 N travelled in a direction that is opposite to that of 90 N, what is the force?

**Solution:**

Let 90 N be F_{1} and 35 N be F_{2}

East to West is a negative direction of travel, therefore F_{1} = – 90 N

35 N travelled in the opposite direction to 90 N,

therefore F_{2} = +35 N

Resultant vector R = – vector F_{1} + vector F_{2}

Resultant vector R = – 90N + 35N = – 55N

### Example 3:

Two vectors of equal magnitude that are pointing in opposite directions will sum to zero. \( \scriptsize \vec{B} \: – \: \scriptsize \vec{D} = 0\)

R = 0.

### Example 4:

In the vector diagram below, find the resultant of the two vectors.

**Solution:**

When two vectors are inclined at 90Â°^{ }(perpendicular) to each other, their resultant is obtained using the Pythagoras theorem.

The resultant of the vectors is represented by the hypotenuse of the triangle in such a way that the direction of the resultant vector is opposite to the direction of the two vectors. The length of the hypotenuse represents the magnitude/size of the resultant vector.

Using Pythagoras,

**(i)**

R^{2}Â = 10^{2} + 12^{2}

R^{2} = 100 + 144

R^{2}Â = 244

R = \( \scriptsize \sqrt{244} \)

= 15.6N

**(ii) **The direction is given by

Î¸ = tan^{-1 }(0.833) = \( \scriptsize 39.8^{\circ} \)

The resultant can also be found by scale drawing.

The resultant vector is a single vector which would have the same effect in magnitude and direction as the original vectors acting together.

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