 Hi and welcome to the session. Let us discuss the following question. Question says, if a unit vector A makes angles pi upon 3 with i vector pi upon 4 with j vector and an acute angle theta with k vector then find theta and hence the components of vector A. First of all let us understand that if it is given that l, m and n are direction cosines of a vector then li plus mj plus nk is equal to cos alpha i plus cos beta j plus cos gamma k is the unit vector in the direction of that vector where alpha, beta and gamma are the angles which the vector makes with x, y and z axis. This is the key idea to solve the given question. Let us now start with the solution. Now we are given that vector A makes an angle pi upon 3 with unit vector i. Now we know vector i is a unit vector along x axis. So we can say cos alpha is equal to cos pi upon 3 is equal to half. Using key idea we get alpha is equal to pi upon 3. We are also given that vector A makes an angle pi upon 4 with unit vector j. Now we know vector j is a unit vector along y axis. So using key idea we get beta is equal to pi upon 4. Now cos beta is equal to cos pi upon 4 which is further equal to 1 upon root 2. Now if vector A makes angles alpha, beta and gamma with x, y and z axis then cos square alpha plus cos square beta plus cos square gamma is equal to 1. We know cos alpha, cos beta and cos gamma are direction cosines of vector A and sum of squares of direction cosines is equal to 1. Or we can say if lmn are direction cosines of a vector then l square plus n square plus n square is equal to 1. Now here l is equal to cos alpha, m is equal to cos beta and n is equal to cos gamma. Substituting these values of lmn in this expression we get this expression. Now we will substitute corresponding values of cos alpha and cos beta. Here we know cos alpha is equal to 1 upon 2 and cos beta is equal to 1 upon root 2. So we can write this implies square of 1 upon 2 plus square of 1 upon root 2 plus cos square gamma is equal to 1. Now this further implies 1 upon 4 plus 1 upon 2 plus cos square gamma is equal to 1. Now adding these two terms we get 3 upon 4. So we can write this expression as 3 upon 4 plus cos square gamma is equal to 1. Now subtracting 3 upon 4 from both the sides of this expression we get cos square gamma is equal to 1 minus 3 upon 4. Now this further implies cos square gamma is equal to 1 upon 4 taking square root on both the sides of this expression we get cos gamma is equal to plus minus 1 upon 2. Now we are given in the question that unit vector a makes an acute angle theta with unit vector k. So we will neglect negative value of cos gamma here and we will substitute theta for gamma and we get cos theta is equal to 1 upon 2. We are given theta is acute. So we will take cos theta is equal to 1 upon 2 and we will neglect negative value that is minus 1 upon 2. Now we know cos 60 degrees is equal to 1 upon 2. Comparing these two equations we get theta is equal to 60 degrees or we can say theta is equal to pi upon 3. Now components of vector a is equal to cos alpha i plus cos beta j plus cos theta k. Now substituting corresponding values of alpha beta and theta here we get vector a is equal to cos pi upon 3 i plus cos pi upon 4 j plus cos pi upon 3 k. Now this further implies vector a is equal to 1 upon 2 i plus 1 upon root 2 j plus 1 upon 2 k. So we get value of theta is equal to pi upon 3 and components of vector a are 1 upon 2, 1 upon root 2 and 1 upon 2. This is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.