Q.Show that $\vec{a} = \dfrac{\hat{i}-\hat{j}}{\sqrt{2}}$ is a unit vector.

Physical quantities fall into two fundamental classes. A scalar quantity is completely specified by a magnitude alone — a number together with a unit — with no reference to direction; mass, time, temperature, density, and distance travelled are all scalars, and they combine using the ordinary rules of algebra. A vector quantity needs both a magnitude AND a direction for its complete description; displacement, velocity, force, and momentum are common examples.

A vector is represented geometrically by a directed line segment (an arrow) drawn to scale, with its length giving the magnitude and its orientation giving the direction; the starting point is called the tail and the arrowhead end is called the head. Several named types of vectors recur throughout physics: the zero (null) vector, which has zero magnitude and an arbitrary direction (e.g. the velocity of a stationary body); the resultant vector, the single vector equivalent to two or more vectors acting together; the negative vector, which has the same magnitude as a given vector but points in exactly the opposite direction; equal vectors, which must share both the same magnitude and the same direction; the position vector, which locates a particle relative to a chosen origin; and the unit vector, a vector of magnitude exactly 1 obtained by dividing any non-zero vector by its own magnitude, $\hat u_M = \vec M/|\vec M|$.

In a rectangular coordinate system, the unit vectors along the x, y, and z axes are denoted $\hat i$, $\hat j$, and $\hat k$ respectively, and every vector can be expressed in terms of these — the foundation for resolving vectors into components and for the algebraic (component) forms of vector addition and multiplication.