Q.If $\vec{v_1} = 3\hat{i}+4\hat{j}+\hat{k}$ and $\vec{v_2} = \hat{i}-\hat{j}-\hat{k}$, determine the magnitude of $\vec{v_1}+\vec{v_2}$.

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.

Both constructions confirm two key algebraic properties of vector addition: it is commutative ($\vec P+\vec Q=\vec Q+\vec P$) and associative ($(\vec A+\vec B)+\vec C=\vec A+(\vec B+\vec C)$), so any number of vectors can be added in any order or grouping without changing the final resultant — the basis for chaining several vectors head-to-tail to find a combined resultant.

Searches for "vector addition triangle law and parallelogram law formula" and "vector addition class 11 physics important questions" are among the most common queries in the Motion in a Plane / Mathematical Tools chapters of the NCERT-aligned CBSE Class 11 Physics curriculum, since these two constructions are foundational to nearly every JEE Main and NEET mechanics question involving vectors. The resultant-magnitude formula $R=\sqrt{P^2+Q^2+2PQ\cos\theta}$ derived here is one of the very first formulas students are expected to apply instantly and correctly in competitive exams.