Question

Vector C is the sum of two vectors A and B and vector D is the cross product of vectors A and B. What is the angle between vectors C and D?
(A) Zero
(B) $ 60^\circ $
(C) $ 90^\circ $
(D) $ 180^\circ $

Answer 1

Hint : Addition of two or more vectors takes place in the same place while the result of the cross product of two vectors lies perpendicular to the plane of these two vectors.

Complete step by step answer
From the question, we know that the relation between vectors C, A and B is $ {\rm{\vec C}} = {\rm{\vec A}} + {\rm{\vec B}} $ and the relation between vectors D, A and B is $ {\rm{\vec D}} = {\rm{\vec A}} \times {\rm{\vec B}} $ .
We now know that the result of the addition of two or more vectors lies in the same plane. So, $ {\rm{\vec C}} $ lies in the same plane as $ {\rm{\vec A}} $ and $ {\rm{\vec B}} $ .
Also, we know that the result of the cross product of two vectors lies in the perpendicular to the plane of these two vectors. So, $ {\rm{\vec D}} $ lies perpendicular to the plane of $ {\rm{\vec A}} $ and $ {\rm{\vec B}} $ .
Since $ {\rm{\vec C}} $ lies in the same plane as of $ {\rm{\vec A}} $ and $ {\rm{\vec B}} $ , so $ {\rm{\vec D}} $ is perpendicular to the $ {\rm{\vec C}} $ .
Hence, the angle between $ {\rm{\vec C}} $ and $ {\rm{\vec D}} $ is equal to $ 90^\circ $ and the option (C) is correct.

Note
Cross product is defined as the vector whose magnitude is equal to the product of the magnitude of two vectors and sine of the angle of between the two vectors and its direction is perpendicular to the plane in which the two vectors lie and it can be obtained by using the right hand rule.
Mathematically, suppose $ {\rm{\vec A}} $ and $ {\rm{\vec B}} $ are two vectors having $ \theta $ angle between them,
 $ {\rm{\vec A}} \times {\rm{\vec B}} = \left| A \right|\left| B \right|\sin \theta $ .