Q.If two sides of a triangle taken in the same order represented by vectors $\hat i+2\hat j$ and $\hat i+\hat k$, then find the length of the third side.

Two vectors acting simultaneously from a common point are added by the Parallelogram Law: if AB=a and AD=b are adjacent sides of a parallelogram, the diagonal AC through the common vertex is a+b. Equivalently, the Triangle Law adds vectors placed tip-to-tail: if AB=a and BC=b, then AC=a+b closes the triangle. Extending the triangle law around several vectors placed tip to tail gives the polygon law, and when the vectors return to the starting point their sum is zero -- which is why, for any triangle ABC, AB+BC+CA=0. Addition is commutative (a+b=b+a) and associative, has the zero vector as identity, and every vector has an additive inverse (its negative). Subtract …