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If vec An d vec B are two vectors and k...

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`

Text Solution

Verified by Experts

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)`
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