Vector Treatment of Forces

Understanding Vectors in the Context of Forces

In the mechanics of forces, a vector is a quantity that has both a magnitude (size) and a direction. Forces are an example of vector quantities, as are displacement, velocity, and acceleration.
Vector quantities are distinct from scalar quantities, which have magnitude only. Examples of scalar quantities include mass, speed, distance, energy, and time.
When dealing with mechanics, always consider whether quantities are vectors or scalars.

Mathematical Representation of Vectors

A vector can be represented in column vector form, where the vertical component is listed above the horizontal component. This convention is often the inverse in physics.
For example, a force of 5N acting to the right and 3N acting upwards could be represented as a column vector: (5 3).
Unit vectors are a useful tool in vector calculations. The unit vector i represents 1 unit in the positive x-direction, while the unit vector j represents 1 unit in the positive y-direction.
Forces can thus be expressed in unit vector notation. For example, a force of 5N to the right and 3N upwards can be written as 5i + 3j.

Vector Addition and Subtraction

When calculating the resultant of multiple forces acting upon a point, the principle of superposition applies. This states that the resultant force is equivalent to the vector sum of the individual forces.
Vector subtraction can be treated as adding a negative vector. For instance, subtracting vector B from vector A is equivalent to adding the negative of vector B to vector A.

Resolving Vectors

It’s sometimes useful to split a vector into components that act in the vertical and horizontal direction. This is known as resolving a vector.
To resolve vectors, you need to understand trigonometry. Remember, the horizontal component of a vector can be found by multiplying its magnitude by the cosine of its angle with the horizontal, while the vertical component is found by multiplying its magnitude by the sine of this angle.
When angles aren’t 90 degrees, resolving vectors can simplify calculations. This approach is especially useful for working with inclined planes or forces that aren’t perpendicular or parallel to the plane.

Resultant Forces and Vector Diagrams

The resultant force is the single force that replaces the effect of several forces acting on a body. It is found by vector addition of all the individual forces.
A force polygon can be drawn to visually represent and calculate resultant forces. In a force diagram, the resultant force is represented by a vector from the start of the first force vector to the end of the last.
The equilibrium condition occurs when the resultant force acting on a body is zero. In this case, the body remains at rest or continues at a constant velocity if already in motion.

Remember, always draw vector diagrams accurately and with correct scale to simplify your calculations and ensure correct results. Recognise the crucial role of vectors in mechanics and practice various vector representations to gain proficiency.