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 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.

For any two vectors vec aa n d vec b , prove that | vec a+ vec b|lt=| vec a|+| vec b| (ii) | vec a- vec b|lt=| vec a|+| vec b| (iii) | vec a- vec b|geq| vec a|-| vec b|

For any three vectors vec a , vec b , vec c , prove that | vec a+ vec b+ vec c|^2=| vec a|^2+| vec b|^2+| vec c|^2+2( vec adot vec b+ vec bdot vec c+ vec cdot vec a)

For any vector vec a\ a n d\ vec b prove that | vec a+ vec b|lt=| vec a|+| vec b|dot

For any two vectors vec a and vec b , show that : ( vec a+ vec b)dot( vec a- vec b)=0, when | vec a|=| 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

Find the angle between two vectors vec a and vec b , if | vec a xx vec b|= vec a . vec b .

If vec a+vec b+vec c=0 , prove that (vec a xx vec b)=(vec b xx vec c)=(vec c xx vec a)

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

If two vectors vec a and vec b are such that | vec a|=2,| vec b|=1 and vec adot vec b=1 , find (3 vec a-5 vec b)dot(2 vec a+7 vec b)dot