Vectors and Scalars Overview

• Vectors and scalars are both types of quantities in physics that are expressed as measurements.
• A scalar quantity is only defined by its magnitude.
• A vector quantity is defined by both its magnitude and its direction.
• Examples of scalar quantities include speed, distance, mass, temperature and energy.
• Examples of vector quantities include displacement, velocity, force and acceleration.

Scalars

• A scalar has only magnitude (size).
• Scalars can be added, subtracted, multiplied, and divided like normal numbers, which is termed as scalar arithmetic.
• Examples of scalar quantities: Mass, distance, speed, time, temperature and energy.
• Note that speed is a scalar. It measures the rate of change of distance but disregards the direction of motion.

Vectors

• Vectors have both magnitude and a direction.
• The direction component of a vector separates it from a scalar quantity.
• Vectors are represented by arrows, where the arrowhead points to the direction and the length signifies the magnitude.
• Vector quantities include: Displacement, velocity, acceleration, and force.
• Displacement, velocity, and acceleration are vectors associated with motion.
• It's important to distinguish between velocity (a vector) and speed (a scalar). Velocity includes the speed of an object and its direction of motion.
• Vectors can be added or subtracted — this is a process called vector addition or vector resolution.

Vector Addition

• Vector addition involves geometrical methods to 'add' vectors known as the "head-to-tail" method or parallelogram method.
• If multiple vector quantities are affecting a single body, the resultant vector can be found using vector addition methods.
• Negative vectors: A vector in the opposite direction is considered ‘negative’.

Scalar and Vector Products

• The dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number.
• The cross product, also known as vector product, outputs a vector which is perpendicular to the vectors being multiplied. The asocioated rules are as per the right-hand rule.

In both the scalar and vector products, consideration must be given to the units used in order to maintain consistency and accuracy.