Q.Prove that the relation $R$ defined on the set $V$ of all vectors by '$\vec a\,R\,\vec b$ if $\vec a=\vec b$' is an equivalence relation on $V$.

A scalar is completely fixed once you know its size — distance, mass, speed, temperature. A vector needs both a magnitude and a direction — force, displacement, velocity. We draw a vector as a directed segment from an initial point (tail) to a terminal point (tip); if the tail is $A$ and the tip is $B$, the vector is written $\vec{AB}$ or, when we don't care about the specific endpoints, a single lower-case letter with an arrow, $\vec a$. Its length is $|\vec a|$.

Special vectors and relations between vectors.

• Co-initial vectors share the same tail; co-terminous vectors share the same tip.
• Vectors are collinear (parallel) if their lines of action are the same or parallel; they are coplanar if they lie in the same plane or in parallel planes.
• Two vectors are equal iff they have the same magnitude and the same direction — they need not share an initial or terminal point.
• The zero vector $\vec 0$ has zero magnitude and an unspecified (arbitrary) direction; its initial and terminal points coincide.
• A unit vector has magnitude $1$. The unit vector along $\vec a$ (for $\vec a\neq\vec 0$) is written $\hat a$, and $\displaystyle \hat a=\frac{\vec a}{|\vec a|}$, so that $\vec a=|\vec a|\,\hat a$. Every direction has exactly one unit vector pointing along it.
• Two parallel vectors are like vectors if they point the same way and unlike vectors if they point opposite ways; a pair that is neither like nor unlike is simply not collinear.

These definitions are the vocabulary the rest of the chapter is built on: algebra of vectors, position vectors, resolution, and the two products all reduce, sooner or later, to comparing magnitudes and directions of vectors defined this way.