Write the direction cosines of the vector r → = 6 i ^ - 2 j ^ + 3 k ^ .

Let r → = 6 i ^ - 2 j ^ + 3 k ^ . Direction cosines are the ratio of the direction ratio in a particular direction and magnitude of the vector. Here, direction ratios of r → are 6 , - 2 , 3 . Magnitude of r → is, r → = 6 2 + - 2 2 + 3 2 = 36 + 4 + 9 = 7 Let the direction cosines are l , m , n in x , y , z directions respectively. So, l = 6 7 , m = - 2 7 , n = 3 7 . Which are the required direction cosines of the vector.

If a → = i ^ + j ^ , b → = j ^ + k ^ and c → = k ^ + i ^ , write unit vectors parallel to a → + b → - 2 c → .

Given, a → = i ^ + j ^ , b → = j ^ + k ^ , c → = k ^ + i ^ Now, a → + b → - 2 c → = i ^ + j ^ + j ^ + k ^ - 2 k ^ - 2 i ^ = - i ^ + 2 j ^ - k ^ Unit vector parallel to a → + b → - 2 c → = ± a → + b → - 2 c → a → + b → - 2 c → = ± - i ^ + 2 j ^ - k ^ - 1 2 + 2 2 + - 1 2 = ± 1 6 i ^ - 2 j ^ + k ^ .

If a → = i ^ + 2 j ^ , b → = j ^ + 2 k ^ , write a unit vector along the vector 3 a → - 2 b → .

Given: a → = i ^ + 2 j ^ , b → = j ^ + 2 k ^ Therefore, 3 a → - 2 b → = 3 i ^ + 2 j ^ - 2 j ^ + 2 k ^ = 3 i ^ + 6 j ^ - 2 j ^ - 4 k ^ = 3 i ^ + 4 j ^ - 4 k ^ 3 a → - 2 b → = 3 2 + 4 2 + - 4 2 = 41 Hence, Unit vector along 3 a → - 2 b → = 3 a → - 2 b → 3 a → - 2 b → = 1 41 3 i ^ + 4 j ^ - 4 k ^ .