Question

If a unit vector is represented by $ 0.5\hat i + 0.8\hat j + c\hat k $ then the value of is $ 'c' $
(A) $ 1 $
(B) $ \sqrt {0.8} $
(C) $ \sqrt {0.11} $
(D) $ \sqrt {0.01} $

Answer 1

Hint :Use the definition of unit vector to find the value of $ 'c' $ .Unit vector is a vector which has a magnitude of $ 1 $ .If $ \vec A $ is a vector then its unit vector will be, $ \hat A = \dfrac{{\vec A}}{{\left| {\vec A} \right|}} $ . Find the magnitude of it and find the required value.

Complete Step By Step Answer:
We know, Unit vector is a vector which has a magnitude of $ 1 $ .If $ \vec A $ is a vector then its unit vector will be, $ \hat A = \dfrac{{\vec A}}{{\left| {\vec A} \right|}} $ .
We have the vector, $ \hat A = 0.5\hat i + 0.8\hat j + c\hat k $ . Since, it is a unit vector the magnitude of it must be unity.
Therefore, $ \left| {\hat A} \right| = 1 $ .
Putting the value of the vector we get,
 $ \sqrt {{{0.5}^2} + {{0.8}^2} + {c^2}} = 1 $
Squaring both sides we get,
 $ {0.5^2} + {0.8^2} + {c^2} = 1 $
Or, $ {c^2} = 1 - ({0.5^2} + {0.8^2}) $
Or, $ {c^2} = 1 - (0.25 + .064) $
Or, $ {c^2} = 1 - 0.89 $
So, we can get, $ {c^2} = 0.11 $
Therefore, taking the square root on both sides we get, $ c = \sqrt {0.11} $ . (Taking the magnitude of $ c $ only)
Hence, the value of $ c $ is $ \sqrt {0.11} $ .
Hence, option ( C) is correct.

Additional Information:
Unit vectors are used to know the direction of a vector. It is widely used in coordinate systems; we use it to point the direction of the axes. In the Cartesian system they are mutually orthogonal. That means they are perpendicular to each other. In the Cartesian system the direction of the unit vectors also means a constant that means they are constant vectors, while in the spherical coordinate system the direction of the unit vectors changes as the vector changes. In a cylindrical coordinate system two of the unit vectors change direction as the vectors. one is only kept constant.
We also have various uses of it like, to find the direction of a surface or to find components of force along any direction etc.


Note :
The unit vector given here can have two orientation here one is $ \hat A = 0.5\hat i + 08\hat j - \sqrt {0.11} \hat k $ or the other, $ \hat A = 0.5\hat i + 08\hat j + \sqrt {0.11} \hat k $ . Both of them reside at the positive side of the X-Y plane.