Mechanics: Vectors

Understanding Vectors in Mechanics

Vectors are quantities that have both a magnitude and a direction. In mechanics, vectors are used to describe physical quantities such as force, velocity, and momentum, all of which have direction.
They are different from scalar quantities, which only have magnitude and no direction, such as speed, mass, or temperature.
Vectors are often represented using arrows, with the length representing the magnitude and the direction of the arrow indicating the direction of the vector.

Two vectors are equal if they have the same magnitude and direction, irrespective of their initial point.
The negative of a vector has the same magnitude but opposite direction.
Vectors maintain their characteristics under parallel translations, meaning that they are not affected by the location in space as long as alignment and magnitude remain the same.

Vectors may be added together or subtracted using geometric methods (such as the triangle or parallelogram rule) or algebraically if the vectors and the angle between them are known.
The addition of vectors obeys commutative law, meaning that the order of addition does not matter: a + b = b + a.
Subtraction of vectors can be simplified as addition of the negative of a vector: a - b = a + (-b).

In mechanics, there are often situations where two vectors are multiplied together. This can be carried out in two ways: Scalar (or dot) product and vector (or cross) product.
Scalar product of two vectors results in a scalar quantity. This is defined as a.b = a b cos θ, where θ is the angle between the two vectors.
Vector product of two vectors results in another vector. This is defined as a × b = a b sin θ n, where θ is the angle between the vectors and n is a unit vector at right angles to a and b.

A vector can be resolved into its components, which means breaking it down into perpendicular parts (usually along the x and y axes). This is particularly useful in mechanics for resolving forces, enabling you to work with single-direction quantities.
The process involves the use of trigonometric functions where the magnitude of the vector serves as the hypotenuse in a right-angled triangle, and the angles provide the vector’s direction.

Be careful about the units used when working with vectors and ensure to convert them appropriately. For instance, if force is measured in Newtons and distance in meters, their scalar product (work done) will be measured in Joules.
Similarly, ensure your answer is in an appropriate unit when calculating vector quantities such as speed, which should be in metres per second (m/s).