For any two vectors ` vec a` and `vec b` prove that `( veca dot vecb)^2le| vec a|^2| vec b|^2` and hence show that `(a_1b_2+a_2b_2+a_3b_3)^2lt=(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)`

For any two vectors vec a and vec b , prove that (vec a xx vec b )^2= |vec a |^2 |vec b|^2 -(vec a. vec b)^2

For any two vectors vec a\ a n d\ vec b prove that | vec axx vec b|^2=| (veca. veca , veca. vecb),(vecb.veca ,vecb.vecb)|

For any two non zero vectors write the value of (| vec a+ vec b|^2+| vec a- vec b|^2)/(| vec a|^2+ | vec b"|^2)

Show that ( vec a- vec b)xx( vec a+ vec b)=2( vec axx vec b)dot

For any two vectors vec aa n d vec b , show that (1+| vec a|^2)(1+| vec b|^2)={"("1+ vec a.vec b")"^2| vec a+ vec b+( vec axx vec b)|^2

Prove the inequality (vec(a) * vec(b) )^(2) le |vec(a)|^(2)|vec(b)|^(2)

Find | vec a- vec b|, if two vector vec a and vec b are such that | vec a|=2,| vec b|=3 and vec adot vec b=4 .

For any two vectors vec a and vec b , prove that: | vec a+ vec b|^2=| vec a|^2+| vec b|^2+2 vec adot vec b , | vec a- vec b|^2=| vec a|^2+| vec b|^2-2 vec adot vec b , | vec a+ vec b|^2+| vec a- vec b|^2=2(| vec a|^2+| vec b|^2) and | vec a+ vec b|^2=| vec a- vec b|^2 iff vec a_|_ vec bdot Interpret the result geometrically.

If | vec a|= a\ a n d\ | vec b|=b, prove that ( vec a/(a^2)- vec b/(b^2))^2=(( vec a- vec b)/(a b))^2

If | vec a|=a and | vec b|=b , prove that ( vec a/(a^2)- vec b/(b^2))^2 = (( vec a- vec b)/(a b))^2 .