You are viewing a free preview of this lesson.
Subscribe to unlock all 10 lessons in this course and every other course on LearningBro.
Physics deals with quantities that describe the world around us — speed, force, mass, temperature, velocity. But not all quantities behave in the same way. Some have only a size (magnitude), while others have both a size and a direction. Understanding this distinction is one of the most fundamental skills in mechanics.
A scalar is a quantity that has magnitude only. It tells you how much of something there is, but nothing about direction.
Common examples of scalars:
| Quantity | SI Unit |
|---|---|
| Mass | kg |
| Speed | m/s |
| Distance | m |
| Time | s |
| Energy | J |
| Temperature | K |
| Power | W |
When you say "the temperature is 293 K" or "the mass is 5.0 kg," direction is irrelevant. A 5.0 kg bag of flour has the same mass regardless of which way it faces.
Scalars are combined using ordinary arithmetic. If you walk 3.0 m and then 4.0 m, the total distance is always 7.0 m, regardless of direction.
A vector is a quantity that has both magnitude and direction. You cannot fully describe a vector without stating both.
Common examples of vectors:
| Quantity | SI Unit |
|---|---|
| Displacement | m |
| Velocity | m/s |
| Acceleration | m/s² |
| Force | N |
| Momentum | kg m/s |
| Weight | N |
Consider the difference between speed and velocity. Speed tells you how fast you are moving (e.g. 5.0 m/s). Velocity tells you how fast and in which direction (e.g. 5.0 m/s due north). Similarly, distance is a scalar (total path length), while displacement is a vector (straight-line distance in a particular direction from start to finish).
Weight vs mass is another important distinction. Mass is a scalar — it is the amount of matter in an object (measured in kg). Weight is a vector — it is the gravitational force acting on the object (measured in N), and it always acts downwards towards the centre of the Earth. Weight = mg, where g = 9.81 m/s².
Vectors are represented by arrows. The length of the arrow represents the magnitude, and the direction of the arrow represents the direction of the vector. In written work, vectors are often shown in bold (e.g. F) or with an arrow above the symbol.
Scalars add algebraically, but vectors must be added geometrically because direction matters.
If you walk 3.0 m east and then 4.0 m north, your total distance is 7.0 m (scalar), but your displacement is only 5.0 m at an angle (vector).
The triangle rule for adding two vectors A and B:
A boat crosses a river at 4.0 m/s due east while the current flows at 3.0 m/s due south. Find the resultant velocity.
The two velocities are perpendicular, so we use Pythagoras:
R = sqrt(4.0² + 3.0²) = sqrt(16 + 9) = sqrt(25) = 5.0 m/s
The angle below the eastward direction:
theta = arctan(3.0 / 4.0) = arctan(0.75) = 36.9°
The resultant velocity is 5.0 m/s at 36.9° south of east (or equivalently, bearing 127°).
When two vectors act from the same point, you can use the parallelogram rule:
This gives the same result as the triangle rule — it is simply a different geometric construction.
Any vector can be split into two perpendicular components, usually horizontal and vertical. This is called resolving a vector.
For a vector F at angle theta to the horizontal:
A force of 50 N acts at 30° above the horizontal. Find the horizontal and vertical components.
Fx = 50 cos(30°) = 50 x 0.866 = 43.3 N (horizontal) Fy = 50 sin(30°) = 50 x 0.500 = 25.0 N (vertical)
This technique is essential throughout mechanics. Whenever forces act at angles, resolve them into components before applying Newton's laws.
An object is in equilibrium when the resultant force acting on it is zero. This means:
If three forces act on an object in equilibrium, they can be represented as a closed triangle when drawn tip-to-tail. This is because the resultant of any two of the forces is equal and opposite to the third.
A 2.0 kg lamp hangs from two cables. Cable A makes an angle of 40° to the horizontal and cable B makes an angle of 60° to the horizontal. Find the tension in each cable.
Weight of lamp: W = mg = 2.0 x 9.81 = 19.6 N (downwards)
Resolving horizontally (taking rightward as positive):
T_A cos(40°) = T_B cos(60°)
Resolving vertically (taking upward as positive):
T_A sin(40°) + T_B sin(60°) = 19.6
From the horizontal equation: T_A = T_B cos(60°) / cos(40°) = T_B x 0.500 / 0.766 = 0.653 T_B
Substituting into the vertical equation:
0.653 T_B x sin(40°) + T_B sin(60°) = 19.6 0.653 T_B x 0.643 + T_B x 0.866 = 19.6 0.420 T_B + 0.866 T_B = 19.6 1.286 T_B = 19.6 T_B = 15.2 N
T_A = 0.653 x 15.2 = 9.9 N