Scalars and Vectors

About this topic

A scalar is described by a number and unit alone, such as mass, time, temperature or speed. A vector also has a direction, such as displacement, velocity, force and acceleration. Vectors are drawn as arrows whose length shows magnitude and whose pointing shows direction. Telling the two apart matters because they follow different rules: scalars add like ordinary numbers, while vectors do not.

Vectors are added tip-to-tail, and the result is the resultant. Perpendicular vectors combine using the Pythagorean theorem, and any vector can be split into x and y components with sine and cosine: Vx = V cos θ and Vy = V sin θ. This splitting is the key to solving most mechanics problems.

Example: walk 6 m east and then 8 m north. The scalar distance is 14 m, but the resultant displacement is the square root of 36 + 64 = 10 m, at about 53 degrees north of east. Pilots, sailors and engineers use exactly this kind of vector addition to plan routes and combine forces on structures.

A common mistake is adding vector magnitudes directly, or forgetting that opposite directions give opposite signs. Always choose a positive direction first, and ask yourself whether each quantity needs a direction before deciding how to handle it.