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 xxvec 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 le |quad vec a|^2|quad vec b|^2 .

For any two vectors vec a and vec b prove that | vec a + vec b | <= | vec a | + | vec b |

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 to vectors vec A and vec B , prove that |vec A xx vec B|^2 = A^2 B^2 - ( vec A. vec B)^2 .

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 and vec b prove that |vec a+vec b|<=|vec a|+|vec b|

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

For any two vectors vec(a) and vec(b) , prove that |vec(a)+vec(b)|le|vec(a)|+|vec(b)| .