Show that the vectors a → = 2 i ^ + 3 j ^ and b → = 6 i ^ - 4 j ^ are at right angles to each other.

Given a → = 2 i ^ + 3 j ^ and b → = 6 i ^ + 4 j ^ if two vectors A → , B → are at right angle to each other. Then their dot product becomes zero. A → . B → = AB cos θ = AB × 0 = 0 We have a → . b → = 2 × 6 + 3 × ( - 4 ) = 0 Therefore, these vectors are at right angle to each other.

Show that the vectors a → = 2 i ^ + 3 j ^ and b → = 6 i ^ - 4 j ^ are at right angles to each other.

Given a → = 2 i ^ + 3 j ^ and b → = 6 i ^ + 4 j ^ if two vectors A → , B → are at right angle to each other. Then their dot product becomes zero. A → . B → = AB cos θ = AB × 0 = 0 We have a → . b → = 2 × 6 + 3 × ( - 4 ) = 0 Therefore, these vectors are at right angle to each other.

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

Given a vector r → = 2 i ^ + j ^ + 3 k ^ To find the direction cosines of the vector one has to divide the corresponding coordinate of the vector by the length of the vector. The coordinates of the unit vector is equal to its direction cosines. The magnitude of the vector r = r 2 x + r 2 y + r 2 z = 2 2 + 1 2 + 3 2 = 14 Therefore, the direction cosines are 2 14 , 1 14 , 3 14

Which vector is to be added with the vectors 3 i ^ + 4 j ^ - 5 k ^ and i ^ - 2 j ^ + 7 k ^ to give a resultant i ^ ?

Given the vector A → = 3 i ^ + 4 j ^ - 5 k ^ B → = i ^ - 2 j ^ + 7 k ^ To have the resultants be unit so that their sum is zero. To have this one should subtract a vector which is the sum of these vectors. Therefore, the vector required C → = ( 3 i ^ + 4 j ^ - 5 k ^ ) + ( i ^ - 2 j ^ + 7 k ^ ) = - 3 i ^ - 2 j ^ - 2 k ^