A line makes angles , and with the positive directions of the coordinate axes. If a = ( ^2 ) i + ( ^2 ) j + ( ^2 ) k and b = i + j + k , then what is a b equal to?

Step-by-step solution

Let , , and be the direction cosines of the given line. We know that the sum of the squares of the direction cosines is 1 : ^2 + ^2 + ^2 = 1 The given vectors are: a = ( ^2 ) i + ( ^2 ) j + ( ^2 ) k b = i + j + k The dot product a b is given by: a b = ( ^2 )(1) + ( ^2 )(1) + ( ^2 )(1) a b = ^2 + ^2 + ^2 Using the identity ^2 = 1 - ^2 , we get: a b = (1 - ^2 ) + (1 - ^2 ) + (1 - ^2 ) a b = 3 - ( ^2 + ^2 + ^2 ) Substituting ^2 + ^2 + ^2 = 1 : a b = 3 - 1 = 2 Answer: 2