1.4 Scalars and vectors

Mass, time, temperature and energy are scalars. Displacement, velocity, acceleration, force and momentum are vectors requiring direction as well as size.

Use scalar arithmetic only for scalars; vector components or geometry are needed when direction changes. A negative signed scalar can represent a component, not a vector by itself.

Speed is the magnitude of velocity; two velocities of 5 m s⁻¹ in opposite directions have equal speeds but a resultant velocity of zero.

A vector is not simply a scalar with an arrow drawn beside it; its direction affects addition and physical effect.

To add vectors, place the tail of one at the head of the other; the resultant joins the first tail to the final head. In components, add corresponding horizontal and vertical parts.

Preserve direction and scale in diagrams, then use Pythagoras or trigonometry only after resolving or forming the correct triangle.

Vectors (3,4) and (−1,2) add to (2,6), whose magnitude is √40.

Adding magnitudes ignores angle; opposite vectors can cancel even when each has a large magnitude.

Resolve a vector into perpendicular components Vx=Vcosθ and Vy=Vsinθ. Recover magnitude with √(Vx²+Vy²) and direction with a quadrant-aware tangent.

Choose axes before resolving, keep signs and use the angle measured from the correct axis. Components are not independent physical vectors unless the coordinate model says so.

A 10 N force at 30° above horizontal has components 10cos30° and 10sin30°.

Swapping sine and cosine changes which component is largest; the reference axis fixes the choice.