Lesson 1, Topic 1
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# Scalars & Vectors

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### Topic Content:

• Meaning of Scalar Quantity
• Meaning of Vector Quantity
• Representation of Vector

### What are Scalar Quantities?

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

### Examples of Scalar Quantities:

Examples of Scalar Quantities are:

• Mass
• Length
• Density
• Time
• Distance
• Temperature
• Speed
• Volume
• Density
• Area
• Current

WithÂ 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  Â  Â  Â  Â  Â  â€¢ V1 + V2 = 10 cm3 + 30 cm3 = 40 cm3
Â  Â  Â  Â  Â  Â  â€¢ T1 + T2 = Â 273 K + 100 K = 373 K
Â  Â  Â  Â  Â  Â  â€¢ A1 + A2 = 200 m2 + 250 m2 = 450 m2

### What are Vector Quantities?

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

### Examples of Vector Quantities:

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 one end pointing in a particular direction. The length of the line represents the magnitude of the vector, for example:

The vector $$\vec{\scriptsize AB}$$ is 10 N in the Eastern direction.

### Example 1.1.1:

If two vectors $$\vec{\scriptsize B}$$ and $$\vec{\scriptsize D}$$ moved in the same direction and are 10 N and 12 N respectively, then, their resultant R is:

R = 22N

### Example 1.1.2:

If Â $$\scriptsize \vec{B}$$ andÂ  $$\scriptsize \vec{D}$$Â 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.Â $$\vec{\scriptsize D}$$

### Example 1.1.3:

Given that 40 N force F1 travels from West to East and 25 N force F2 travels in the opposite direction to that of 40 N force. Find the resultant of the two vectors.

Solution:

Resultant Vector R = vector F1 + ( – vector F2)
Resultant vector R = vector F1 â€“ vector F2
Resultant vector R = 40 N â€“ 25 N = 15 N

### Example 1.1.4:

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 F1 and 35 N be F2
East to West is a negative direction of travel, therefore F1 = – 90 N
35 N travelled in the opposite direction to 90 N,Â
therefore F2 = + 35 N

Resultant vector R = – vector F1 + vector F2
Resultant vector R = – 90N + 35N = – 55N

### Example 1.1.5:

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 1.1.6:

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

Solution:

(i) 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)

R2 = 102 + 122

R2 = 100 + 144

R2 = 244

R = $$\scriptsize \sqrt{244}$$

Â Â Â = 15.6 N

(ii) The direction is given by

$$\scriptsize tan \: \theta = \normalsize \frac{opp}{adj} = \frac{10}{12}$$

$$\scriptsize tan \: \theta = 0.833$$

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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