 Good morning friends and Pooja and today we will discuss the following question. Let vector a and vector b be two unit vectors and Theta is the angle between them. Then vector a plus vector b is a unit vector. If a Theta is equal to pi by 4, b, theta is equal to pi by 3, c, theta is equal to pi by 2, d, theta is equal to 2 pi by 3. Let us now begin with the solution. Now we are given that vector a and vector b are unit vectors. So since vector a and vector b are unit vectors, therefore we have mod of vector a is equal to mod of vector b is equal to 1. Now we are also given that the angle between the vectors vector a and vector b is theta. So we have cos theta is equal to vector a dot vector b upon mod of vector a into mod of vector b. This is given to us. Now since mod of vector a is equal to mod of vector b is equal to 1, therefore we get cos theta is equal to vector a dot vector b and this is since mod of vector a is equal to mod of vector b is equal to 1. We mark this as equation 1. Now we have to find the value of theta for which vector a plus vector b is a unit vector. So now vector a plus vector b is a unit vector therefore we have mod of vector a plus vector b is equal to 1. Squaring both the sides we get mod of vector a plus vector b whole square is equal to 1. Now this implies mod of vector a square plus 2 into vector a dot vector b plus mod of vector b square is equal to 1. Since we know that a plus b whole square is equal to a square plus 2 a b plus b square. Now we are given that mod of vector a and mod of vector b is equal to 1. So we get this implies 1 square which is equal to 1 plus 2 into vector a dot vector b plus 1 is equal to 1 and this implies 2 into vector a dot vector b is equal to 1 minus 2 which further implies 2 into vector a dot vector b is equal to minus 1. This further implies vector a dot vector b is equal to minus 1 upon 2. Now from equation one we know that cos theta is equal to vector a dot vector b. So we get cos theta is equal to minus 1 by 2 and this is from equation one. Now cos theta is equal to minus 1 upon 2 when theta is equal to 2 pi by 3. So we get this implies cos theta is equal to cos 2 pi by 3 and this further implies theta is equal to 2 pi by 3 and this is same as option d. So vector a plus vector b is the unit vector if theta is equal to 2 pi by 3. So our correct option is d. This is our answer. Hope you have understood the solution. Bye and take care.