For what value of c the vector  will be perpendicular to:
(i) Z-axis    (ii) Y-axis?
(i) Vector  is zero.
i.e. Â Â Â Â Â Â Â Â Â Â Â Â
         Â
          Â
The equation is satisfied for all values of c.
Therefore  is always perpendicular to Z-axis whatever the value of c.
(ii) Similarly    Â
          Â
              Â
                  Â
∴   is perpendicular to Y-axis if c = 0.
A particle moves from point (2, 3, 5) to (8, -4, 2) when a force of  unit is applied on it. Find the work done by the force.
Given, a particle moves from point P(2,3,5) to Q(8,-4,2).Â
Force applied on th particle, F =Â Â
Therefore, the displacement undergone by the particle is,Â
       Â
         Â
Therefore work done by force is,Â
 Â
         J
Â
Find the unit vector in direction of  whereÂ
Here, Â Â
Therefore the magnitude of vectorÂ
Â
   Â
Now the unit vector in direction of  is,Â
  Â
What is the angle between the vectors  and ?
Given that,
     Â
and   Â
Let  be the angle between the given two vectors.
We know, Â
∴        Â
Now      Â
            Â
Magnitude is given by,
Â
 Thus, angle between the vectors,
Â
          Â