Q.Distinguish between scalar and vector quantities, giving two examples of each. Why can ordinary algebraic addition be used for scalars but not for vectors in general?

[!TLDR] Scalars need only magnitude; vectors need magnitude and direction, so they need vector algebra, not ordinary algebra. [!ANSWER] Scalars (e.g. mass, time) add by ordinary arithmetic; vectors (e.g. displacement, force) must add by the triangle/parallelogram law because direction has to be accounted for.

A scalar quantity is completely described by a single number and a unit — for example, a mass of 5 kg or a time of 10 s. Two scalars of the same kind combine by simple arithmetic: 5 kg + 3 kg = 8 kg, unconditionally. A vector quantity, in contrast, needs both a magnitude and a direction — for example, a displacement of 5 m due east, or a force of 10 N at $30^\circ$ to the horizontal. Two vectors of the same kind cannot, in general, be combined by simply adding their magnitudes, because the result depends on the angle between them as well: two 5 N forces pointing in the same direction combine to 10 N, but the same two forces pointing in opposite directions combine to 0 N, and at any other angle the resultant lies somewhere in between. This angle-dependence is why vectors need their own algebra (the triangle law, parallelogram law, and the component method), rather than ordinary arithmetic. Examples of scalars: mass, distance, time, speed, temperature, energy. Examples of vectors: displacement, velocity, acceleration, force, momentum. [!ANSWER] Scalars (e.g. mass, time) add by ordinary arithmetic; vectors (e.g. displacement, force) must add by the triangle/parallelogram law because direction has to be accounted for.

Define both terms with examples, then explain why direction-dependence forces vectors to use a different addition rule.

• Calling speed a vector (speed is the scalar magnitude of the vector velocity).\n- Thinking any quantity with a '+' or '-' sign is automatically a vector.