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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.

1. ii only
2. i and ii
3. iii only
4. iv only
5. iv and v

Let n =1 then,

if a = { 2 } , b = { 1 }

< 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.