Question

Which of the following is not true?
I. $a\cdot b$ is a real number.
II. $a\cdot b=0\iff a\Vert b;a\ne 0,b\ne 0$
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., $a\cdot b=0\iff a\perp b$.
(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.