Scalars & Vectors

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:

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

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:

Example 1.1.2:

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 – vector F₂
Resultant vector R = 40 N – 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, 
therefore F₂ = + 35 N

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

Example 1.1.5:

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 90° (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 = \( \scriptsize \sqrt{244} \)

   = 15.6 N

(ii) The direction is given by

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

\( \scriptsize tan \: \theta = 0.833\)

θ = 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.