= | | λ , provided ≠ i.e. is not a null vector, then | λ | =| λ || | = So, λ represents the unit vector in the direction of . We write it as ˆ a = | | r a A Note For any scalar k , = . .
. Components of a vector Let us take the points A( , , ), B( , , ) and C( , , ) on the x -axis, y -axis and z -axis, respectively. Then, clearly | |= , | |= and | |= The vectors , each having magnitude , are called unit vectors along the axes OX, OY and OZ, respectively, and denoted by ˆ ˆ , and i j k , respectively (Fig . ).
Now, consider the position vector of a point P( x , y , z ) as in Fig . . Let P be the foot of the perpendicular from P on the plane XOY. We, thus, see that P P is parallel to z -axis.
As ˆ ˆ , and i j k are the unit vectors along the x , y and z -axes, respectively, and by the definition of the coordinates of P, we have . Similarly, Fig . Fig . Therefore, it follows that Hence, the position vector of P with reference to O is given by This form of any vector is called its component form.
Here, x , y and z are called as the scalar components of , and xi yj zk are called the vector components of along the respective axes. Sometimes x , y and z are also termed as rectangular components . The length of any vector = xi yj zk , is readily determined by applying the Pythagoras theorem twice. We note that in the right angle triangle OQP (Fig .
) and in the right angle triangle OP P, we have Hence, the length of any vector = ˆ+ xi yj zk is given by | | = If are any two vectors given in