Scalars and Vectors – SS1 Physics Lesson Note (First Term)

Scalar quantities have magnitude only, while vector quantities have both magnitude and direction. Vectors can be added to give a single vector called the resultant, or split into two components.

Behavioural objectives

1. Distinguish between scalar and vector quantities and give examples of each.
2. Represent a vector graphically.
3. Find the resultant of two vectors acting in the same direction, in opposite directions and at 90° to each other.
4. State the parallelogram law of vector addition.
5. Resolve a vector into horizontal and vertical components.
6. Describe how to find the resultant of more than two vectors.

What are scalar and vector quantities?

A scalar quantity has magnitude only, while a vector quantity has both magnitude and direction.

• Physical quantity: any property of a body or event that can be measured.
• Magnitude: the size of a quantity, for example 5 N or 20 m.
• Direction: the line along which a quantity acts, for example north or 30° above the ground.
• Scalar quantity: a quantity with magnitude only.
• Vector quantity: a quantity with both magnitude and direction.

What are examples of scalar quantities?

• Distance: the total length of path covered, such as 5 km.
• Speed: distance covered per unit time.
• Temperature: how hot or cold a body is.
• Volume: the space a body occupies.
• Mass: the amount of matter in a body.
• Work, energy and power: each has size only, with no direction.
• Time: a size only, measured in seconds.

What are examples of vector quantities?

• Force: a push or pull, always acting in some direction.
• Weight: the force of gravity on a body, acting downwards.
• Displacement: distance in a stated direction.
• Velocity: speed in a stated direction.
• Acceleration: rate of change of velocity.
• Momentum: mass multiplied by velocity.
• Fields: magnetic field, electric field and gravitational field.

What is the difference between scalar and vector quantities?

• Scalar: described fully by a number and a unit, for example a mass of 50 kg.
• Vector: needs a number, a unit and a direction, for example a force of 50 N upwards.

How do distance and displacement, speed and velocity differ?

How is a vector represented?

A vector is represented by a straight line with an arrowhead, drawn to scale.

• Line: its length shows the magnitude of the quantity.
• Arrowhead: it shows the direction of the vector.
• Scale: a fixed length stands for a fixed size, for example 1 cm for 10 N.

What are the steps to draw a vector?

1. Choose a suitable scale, for example 1 cm represents 10 N.
2. Convert the magnitude to a length. A 40 N force becomes a 4 cm line.
3. Draw the line in the stated direction, using a protractor for the angle.
4. Put an arrowhead at the end of the line.

How are vectors added and subtracted?

Vectors are added by finding their resultant, which must allow for direction.

• Resultant: the single vector that has the same effect as two or more vectors acting together.
• Subtraction: to find A − B, reverse the direction of B and add it to A.

How do we add vectors in the same direction?

• Rule: add the magnitudes (the algebraic sum).
• Example: forces Y = 3 N and X = 4 N in the same direction give a resultant of 7 N.

How do we add vectors in opposite directions?

• Rule: subtract the smaller magnitude from the larger.
• Direction: the resultant acts in the direction of the larger vector.
• Example: Y = 3 N and X = 4 N in opposite directions give a resultant of 1 N.

How do we add vectors inclined at 90° to each other?

• Rule: use Pythagoras’ theorem for the magnitude.
• Direction: use the tangent ratio for the angle.

Worked example: X = 4 N and Y = 3 N at 90°

1. R² = X² + Y²
2. R² = 4² + 3²
3. R² = 16 + 9 = 25
4. R = √25 = 5 N
5. tan θ = Y/X = 3/4 = 0.75
6. θ = tan^-1(0.75) = 36.9°

What is the resultant of 3 N and 4 N in the three cases?

How do we add vectors inclined at other angles?

Vectors at any angle other than 0°, 90° or 180° are added with the parallelogram law of vector addition.

• When it applies: the angle between the vectors is less than 90° or more than 90°.
• Method: use the parallelogram law of vector addition.
• Parallelogram law: if two vectors are represented in magnitude and direction by adjacent sides of a parallelogram, the resultant is represented in magnitude and direction by the diagonal drawn from the common point.

Steps for the graphical method

1. Choose a scale and draw both vectors from a common point at the given angle.
2. Complete the parallelogram by drawing lines parallel to each vector.
3. Draw the diagonal from the common point.
4. Measure the diagonal and convert its length to a magnitude using the scale.
5. Measure the angle of the diagonal with a protractor to get the direction.

What is the resolution of vectors?

Resolution is the splitting of a single vector into two vectors, called components, that together have the same effect.

• Method: the vector F is the diagonal of a parallelogram. Its components are the adjacent sides.
• Usual choice: two components at right angles, one horizontal and one vertical.

What are the formulae for the components of a force?

For a force F at angle θ to the horizontal, the components are F cos θ and F sin θ.

• Vertical component: sin θ = y/F, so y = F sin θ.
• Horizontal component: cos θ = x/F, so x = F cos θ.
• Direction of F: tan θ = y/x, so θ = tan^-1(y/x).
• Angles above 90°: cos θ is negative, so the horizontal component points the other way.

Worked example: a 100 N force at 30° to the ground

1. Horizontal component: x = 100 cos 30° = 86.6 N.
2. Vertical component: y = 100 sin 30° = 50 N.

How do we find the resultant of more than two vectors?

The resultant of more than two vectors is found by resolving each vector and adding the components.

What are the steps?

1. Draw two perpendicular axes through the point where the forces act.
2. Resolve each vector into a horizontal and a vertical component.
3. Give each component a sign. Components pointing left or down are negative.
4. Add all the horizontal components to get X.
5. Add all the vertical components to get Y.
6. Find the magnitude: R = √(X² + Y²).
7. Find the direction α from tan α = Y/X.

Example: four forces acting on a body

• X: X = F₁ cos θ₁ + (-F₂ cos θ₂) + (-F₃ cos θ₃) + F₄ cos θ₄
• Y: Y = F₁ sin θ₁ + F₂ sin θ₂ + (-F₃ sin θ₃) + (-F₄ sin θ₄)
• Magnitude: R² = X² + Y², so R = √(X² + Y²).
• Direction: tan α = Y/X.

Worked example: forces already along the axes

Four forces act on a body: 6 N to the right, 2 N to the left, 5 N upwards and 1 N downwards.

1. X = 6 + (-2) = 4 N.
2. Y = 5 + (-1) = 4 N.
3. R = √(4² + 4²) = √32 = 5.7 N.
4. tan α = 4/4 = 1, so α = 45° above the horizontal.

Evaluation

1. Give four examples of scalar quantities and three examples of vector quantities.
2. Two forces of 3N and 4N act on a body. What is the resultant when they act (i) in the same direction, (ii) in opposite directions, (iii) at 90° to each other?
3. State the parallelogram law of vector addition.
4. Write the expressions for the horizontal and vertical components of a force F inclined at an angle θ.

Assignment

Answer the following in your exercise book.

1. Define vector.
2. What is the difference between a scalar and a vector?
3. Find the vertical and horizontal components of a 500N force when it is inclined at (i) 60°, (ii) 90°, (iii) 150° to the ground level.