Prove Cauchy- Schwarz inequality `|quad vec a* vec b|<|quad vec a||quad vec b|`.

If (vec a+ vec b)* (vec a- vec b)=0 , show that |quad vec a|=|quad vec b| .

If vec a and vec b are two vectors such that |quad vec a + vec b|= |quad vec a| , then prove that 2 vec a+ vec b is perpendicular to the vector vec b .

Prove that two proper vectors vec a and vec b are the right angles iff |quad vec a+ vec b|^2 =|quad vec a|^2+ |quad vec b|^2 .

If vec a and vec b are two vectors such that |quad vec a|=2 , |quad vec b|=1 and vec a. vec b=1 , then find the value of (3 vec a - 5 vec b).(2 vec a+7 vec b) .

If vec a and vec b are any two vectors, then prove that |quad vec a+ vec b|^2 + |quad vec a- vec b|^2= 2|quad vec a|^2 + 2|quad vec b|^2 .

If vec a and vec b are two vectors such that |quad vec a| =10 , |quad vec b|=15 and vec a.vec b=75 sqrt 2 , find the angle between vec a and vec b .

If vec c=3vec a+ 4vec b and 2vec c=vec a-vec 3b then show that vec c and vec a are like vectors and |quad vec c|>|quad vec a| .

If |quad vec a|=1 , |quad vec b|=1 and |quad vec a+ vec b|=1 , prove that | vec a- vec b|= sqrt 3 .

If vec a is perpendicular to vec b and vec r is non-zero vector such that p vec r+( vec rdot vec a) vec b= vec c , then vec r= vec c/p-(( vec adot vec c) vec b)/(p^2) (b) vec a/p-(( vec cdot vec b) vec a)/(p^2) vec a/p-(( vec adot vec b) vec c)/(p^2) (d) vec c/(p^2)-(( vec adot vec c) vec b)/p

If vec a, vec b, vec c are three vectors such that |quad vec a|=5,| , quad vec b|=12 and |quad vec c|=13 and if vec a+ vec b+ vec c= 0 , find the value of vec a* vec b + vec b* vec c + vec c* vec a .