### Topic Content:

- Meaning of Scalar QuantityA scalar quantity is a quantity that does not depend on direction. It can also be defined as a quantity with magnitude only. Examples of Scalar Quantities include area, density, distance,... More
- Addition of Scalar Quantities
- Meaning of Vector QuantityVector quantities are quantities with magnitude and direction. Examples of vector quantities include displacement, velocity, position, force, and torque. More
- Representation of Vector
- Addition of Vector Quantities

### 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
- DensityDensity is the measurement of how tightly a material is packed together i.e. how closely the particles are packed in the material. The tighter the material is packed the more its... More
- Time
- Distance
- Temperature
- Speed
- Volume
- Density
- Area
- Current

### Addition of Scalar Quantities:

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€¢ V_{1} + V_{2} = 10 cm^{3 }+ 30 cm^{3 }= 40 cm^{3}

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

Â Â Â Â Â Â â€¢ A_{1} + A_{2} = 200 m^{2 }+ 250 m^{2} = 450 m^{2}

### 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.

### Addition of Vector Quantities:

### 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 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 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 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,Â

therefore F_{2} = + 35 N

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

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)**

R^{2} = 10^{2} + 12^{2}

R^{2} = 100 + 144

R^{2} = 244

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

Â Â Â = 15.6 N

**(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.

This was so cool it actually matched with what I learnt at school

This is fantastic,it just like I am in class again

This is awesome it’s helps for quick understanding

I so enjoy this website

This is lovely! A good progression from easy to complex

I really enjoyed the note. It explicit and the content is valid. Thanks for also making it free.

I love this