Question

If ` vec An d vec B` are two vectors and `k` any scalar quantity greater than zero, then prove that `| vec A+ vec B|^2lt=(1+k)| vec A|^2+(1+1/k)| vec B|^2dot`

Answer 1

We know, `(1+k)|A|^(2)(1+(1)/(k))|B|^(2)`
`=|A|^(2)+k|A|^(2)+|B|^(2)+(1)/(k)|B|^(2)` . . . (i)
Also, `k|A|^(2)+(1)/(k)|B|^(2)ge2(k|A|^(2)*(1)/(k)|B|^(2))^((1)/(2))=1|A|*|B|` . . . (ii)
(since, arithmetic mean `ge` Geometric mean)
So, `(1+k)|A|^(2)+(1+(1)/(k))|B|^(2) ge |A|^(2)+|B|^(2)+2|A|*|B|`
`=(|A|+|B|)^(2)` [using eqs. (i) and (ii)]
And also, `|A|+|B| ge |A+B|`
hence, `(1+k)|A|^(2)+(1+(1)/(k))|B|^(2)ge|A+B|^(2)`.