Find the unit vector in the direction of the vector r → 1 - r → 2 , where r 1 → = i ^ + 2 j ^ + k ^ and r 2 → = 3 i ^ + j ^ - 5 k ^ .

Given vectors are: r 1 → = i ^ + 2 j ^ + k ^ and r 2 → = 3 i ^ + j ^ - 5 k ^ Now, r 1 → - r 2 → = i ^ + 2 j ^ + k ^ - 3 i ^ + j ^ - 5 k ^ = - 2 i ^ + j ^ + 6 k ^ We know that, the unit vector in the direction of the vector n → is n ^ = n → n → Therefore, the unit vector in the direction of the vector - 2 i ^ + j ^ + 6 k ^ = - 2 i ^ + j ^ + 6 k ^ | - 2 i ^ + j ^ + 6 k ^ | = - 2 i ^ + j ^ + 6 k ^ 4 + 1 + 36 = - 2 i ^ + j ^ + 6 k ^ 41 = - 2 41 i ^ + 1 41 j ^ + 6 41 k ^ , which is the required unit vector in the direction of the vector - 2 i ^ + j ^ + 6 k ^ .

Find the unit vector parallel to the sum of the vectors a → = 2 i ^ + 4 j ^ - 5 k ^ and b → = i ^ + 2 j ^ + 3 k ^ . Also find its direction cosines.

Given vectors are: a → = 2 i ^ + 4 j ^ - 5 k ^ and b → = i ^ + 2 j ^ + 3 k ^ Now, then sum of the given vectors is r → = a → + b → = 2 i ^ + 4 j ^ - 5 k ^ + i ^ + 2 j ^ + 3 k ^ = 3 i ^ + 6 j ^ - 2 k ^ Hence, a unit vector parallel to the sum of the vectors a → = 2 i ^ + 4 j ^ - 5 k ^ and b → = i ^ + 2 j ^ + 3 k ^ i.e., r ^ = r → | r → | = 3 i ^ + 6 j ^ - 2 k ^ 3 i ^ + 6 j ^ - 2 k ^ = 3 i ^ + 6 j ^ - 2 k ^ 9 + 36 + 4 = 3 i ^ + 6 j ^ - 2 k ^ 7 = 3 7 i ^ + 6 7 j ^ - 2 7 k ^ Thus, the required direction cosines are: 3 7 , 6 7 , - 2 7 .

The position vectors of the points A , B , C and D are 4 i ^ + 3 j ^ - k ^ , 5 i ^ + 2 j ^ + 2 k ^ , 2 i ^ - 2 j ^ - 3 k ^ and 4 i ^ - 4 j ^ + 3 k ^ . Show that A B → and C D → are parallel.

Let a → , b → , c → and d → are the position vectors of the points A , B , C and D . So, a → = 4 i ^ + 3 j ^ - k ^ , b → = 5 i ^ + 2 j ^ + 2 k ^ , c → = 2 i ^ - 2 j ^ - 3 k ^ and d → = 4 i ^ - 4 j ^ + 3 k ^ Now, A B → = O B → - O A → = b → - a → = 5 i ^ + 2 j ^ + 2 k ^ - 4 i ^ + 3 j ^ - k ^ = i ^ - j ^ + 3 k ^ and C D → = O D → - O C → = d → - c → = 4 i ^ - 4 j ^ + 3 k ^ - 2 i ^ - 2 j ^ - 3 k ^ = 2 i ^ - 2 j ^ + 6 k ^ So, clearly we have, C D → = 2 A B → in other words, the coefficients of i ^ , j ^ , k ^ components are proportional to each other. Thus, by definition A B → and C D → are parallel.