TIFR CSE 2017 | Part A | Question: 2

Question

For vectors $x, \: y$ in $\mathbb{R}^n$, define the inner product $\langle x, y \rangle = \Sigma^n_{i=1} x_iy_i$, and the length of $x$ to be $\| x \| = \sqrt{\langle x, x \rangle}$. Let $a, \: b$ be two vectors in $\mathbb{R} ^n$ so that $\| b \| =1$. Consider the following statements:

1. $\langle a, b \rangle \leq \| b \|$
2. $\langle a, b \rangle \leq \| a \|$
3. $\langle a, b \rangle = \| a \| \| b \|$
4. $\langle a, b \rangle \geq \| b \|$
5. $\langle a, b \rangle \geq \| a \|$

Which of the above statements must be TRUE of $a, \: b$? Choose from the following options.

• A. ii only
• B. i and ii
• C. iii only
• D. iv only
• E. iv and v

Comments

@Bikram sir please explain in a better way 

Answer 1

Let $n =1$ then,

if $a$ $=$ $\left \{ 2 \right \}$, $b$ $=$ $\left \{ 1 \right \}$

< $a,b$ > $=$ $2$

$||$ $b$ $||$ $=$ $1$

Hence, i is incorrect.

Let a = {-1} , b ={1}

$< a,b > = -1$

$||$ $a$ $||$ $||$ $b$ $||$ $=$ $1$ x $1$ = $1$

Hence, iii and iv is incorrect, if iv is incorrect $d$ and $e$ both can't be right.

 So, by elimination $(a)$ is correct.

Answer 2

Note that  $\vec{a} \cdot \hat{b} = \left \| a \right \| \left ( \hat{a} \cdot \hat{b}\right ) \;\;and \;\;\; \hat{a} \cdot \hat{b}$ is dot product of unit vectors, which means that

It is component of one unit vector along the other, which will always be between -1 and 1 .

Hence,

$ \;\;\;\;-1 \leqslant \hat{a} \cdot \hat{b} \leqslant 1$

 $\Rightarrow -\left \| \vec{a} \right \| \leqslant \vec{a} \cdot \hat{b} \leqslant \left \| \vec{a} \right \|$

$\Rightarrow -\left \| a \right \| \leqslant \left \langle a,b \right \rangle \leqslant \left \| a \right \|$ (In our question)

 

Rest all of the other options can't be implied with the limited information provided.

What about $ \left \langle a,b \right \rangle \leqslant \left \| b \right \|$ ?

It is obviously incorrect as $\left \| b \right \|$ is 1, and we don't know $\left \| a \right \|$, to be able to comment on $ \left \langle a,b \right \rangle \leqslant \left \| b \right \|$

So, answer is option A