For any two vectors a and b, which of the following statements is always true ?

Question

For any two vectors \(\vec a\) and \(\vec b\), which of the following statements is always true ?

(A) \(\vec a .\vec b\ge |\vec a| |\vec b|\)

(B) \(\vec a .\vec b = |\vec a| |\vec b|\)

(C) \(\vec a .\vec b \le |\vec a| |\vec b|\)

(D) \(\vec a .\vec b< |\vec a| |\vec b|\)

Answer 1

Correct option is (C) \(\vec a .\vec b \le |\vec a| |\vec b|\) 

Let us assume \(\vec a \ne \vec 0\) and \(\vec b \ne \vec 0\)

\(\vec a. \vec b = |\vec a| |\vec b| \cos \theta\)

Taking modulus on both sides then we get,

\(|\vec a. \vec b| = |\vec a| |\vec b| |\cos \theta|\)

\(|\cos \theta| = \frac{|\vec a.\vec b|}{|\vec a||\vec b|}\)

As we know that \(0 \le |\cos \theta| \le 1\)

\(\frac{|\vec a.\vec b|}{|\vec a||\vec b|}\le 1\)

Therefore, \(|\vec a .\vec b| \le |\vec a| |\vec b|\)