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 xxvec b)^(2)=|vec a|^(2)|vec b|^(2)-(vec a*vec b)^(2)

Prove by vector method that (a_1b_1+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 to vectors vec A and vec B , prove that |vec A xx vec B|^2 = A^2 B^2 - ( vec A. vec B)^2 .

Prove that (vec a * vec b) ^ (2) <= vec a ^ (2) * vec b ^ (2)

(vec a xxvec b) ^ (2) + (vec a * vec b) ^ (2) = | vec a | ^ (2) | vec b | ^ (2)

Prove that |vec a + vec b| = sqrt (|vec a|^2 + |vec b|^2 + 2 vec a * vec b) .

Find |vec a-vec b|, if two vectors vec a and vec b are such that |vec a|=2, and |vec b|=3 and |vec a*vec b|=4

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

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)

Let veca = a_1hati + a_2hatj + a_3hatk, vecb = b_1hati + b_2hatj+ b_3hatk and vecc = c_1hati + c_2hatj + c_3hatk be three non zero vectors such that |vecc| =1 angle between veca and vecb is pi/4 and vecc is perpendicular to veca and vecb then |[a_1, b_1, c_1], [a_2, b_2, c_2], [a_3, b_3, c_3]|^2= lamda(a_1 ^2 +a_2 ^2 + a_3 ^2)(b_1 ^2 + b_2^2+b_3^2) where lamda is equal to (A) 1/2 (B) 1/4 (C) 1 (D) 2