Q.Using the triangle law of vector addition, show that vector addition is both commutative ($\vec A + \vec B = \vec B + \vec A$) and associative ($(\vec A + \vec B) + \vec C = \vec A + (\vec B + \vec C)$).

Only vectors describing the SAME physical quantity may be combined by addition or subtraction — two forces can be added, but a force cannot be added to a velocity. When vectors act along the same straight line, combining them is simple: parallel vectors add their magnitudes directly, while anti-parallel (opposite-direction) vectors combine by SUBTRACTING magnitudes, with the resultant pointing in the direction of the larger vector.

When two vectors do not lie along the same line, their resultant is found using one of two equivalent geometric constructions. The triangle law states that if two vectors are represented, in order, by the two sides of a triangle, their resultant is represented by the third side, drawn from the start of the first vector to the end of the second. The parallelogram law states that if two vectors (with tails at a common point) are represented by two adjacent sides of a parallelogram, their resultant is represented by the diagonal drawn from that same point; this construction, via the Pythagoras theorem, gives the resultant's magnitude as $R=\sqrt{P^2+Q^2+2PQ\cos\theta}$ and its direction via $\tan\alpha = Q\sin\theta/(P+Q\cos\theta)$, where $\theta$ is the angle between the two vectors. …