Note that two vectors are equal if and only if their corresponding components are equal. Thus, the given vectors will be equal if and only if x = , y = , z = Example Let . Is ? Are the vectors equal?
Solution We have So, . But, the two vectors are not equal since their corresponding components are distinct. Example Find unit vector in the direction of vector Solution The unit vector in the direction of a vector is given by Now Therefore ( Example Find a vector in the direction of vector that has magnitude units. Solution The unit vector in the direction of the given vector is ) Therefore, the vector having magnitude equal to and in the direction of is a ∧ = ∧ ∧ Example Find the unit vector in the direction of the sum of the vectors, Solution The sum of the given vectors is ( ) + − Thus, the required unit vector is ( ) Example Write the direction ratio’s of the vector and hence calculate its direction cosines.
Solution Note that the direction ratio’s a , b , c of a vector are just the respective components x , y and z of the vector. So, for the given vector, we have a = , b = and c = – . Further, if l , m and n are the direction cosines of the given vector, then Thus, the direction cosines are ,– . .
Vector joining two points If P ( x , y , z ) and P ( x , y , z ) are any two points, then the vector joining P and P is the vector (Fig . ). Joining the points P and P with the origin O, and applying triangle law, from the triangle