If `vecA and vecB` are two vectors and k any scalar quantity greater than zero, then prove that `|vecA+ vecB|^(2) le (1+k)|vecA|^(2)+ (1+ (1)/(k))|vecB|^(2)`

Explanation

We know :
`" "(1+k) |vecA|^(2) + (1+(1)/(k))|vecB|^(2)`
`" "=|vecA|^(2) + k|vecA|^(2)+ |vecB|^(2)+ (1)/(k)|vecB|^(2)" "`(i)
Also,
`" "k|vecA|^(2) + (1)/(k)|vecB|^(2) ge 2(k+ |vecA|^(2) *(1)/(k)|vecB|^(2))^(1//2)`
`" "=2|vecA|*|vecB|" "`(ii)
(Since arithmetic mean `ge` geometric mean)
`therefore " "(1+k)|vecA|^(2) + (1+(1)/(k))|vecB|^(2) ge |vecA|^(2) + |vecB|^(2) + 2|vecA|*|vecB|= (|vecA|+|vecB|)^(2)` [Using (i) and (ii)]
And also `|vecA|+ |vecB| ge |vecA+ vecB|`
Hence,`(1+k)|vecA|^(2) +(1+ (1)/(k))^(2) |vecB|^(2) ge |vecA+vecB|^(2)`