Question

Which of the following is not true?
I. $a \cdot b$ is a real number.
II. $a \cdot b =0 \Leftrightarrow a \| b ; a \neq 0, b \neq 0$
III. $a \cdot a =| a |^2$
IV. $a \cdot(-a)=|a|^2$

• A. II and IV are not true
• B. II and III are not true
• C. Only III is not true
• D. Only I is not true

Answer 1

Answer: (A)

If $a$ and $b$ are two vector then $a \cdot b =| a || b | \cos \theta$. By this we can conclude that.
(i) $a \cdot b$ is a real number.
(ii) If $a$ and $b$ be two non zero vectors, then $a \cdot b=0$ if and only if $a$ and $b$ are perpendicular to each other, i.e., $a \cdot b=0 \Leftrightarrow a \perp b$.
(iii) If $\theta=0$, then $a \cdot b=|a||b|$. In particular, $a \cdot a=|a|^2$, as $\theta$ in this case is 0 .
(iv) If $\theta=\pi$, then $a \cdot b=-|a||b|$ in particular, $a \cdot|a|=-|a|^2$ as $\theta$ in this case is $\pi$.
So, we see that option (a) is correct.