 Hello and welcome to the session my name is Mansi and I am going to help you with the following question. The question says show that the points minus 2,3,5,1,2,3 and 7,0, minus 1 are co-linear. Let us start with the solution to this question. First of all suppose we have 3 points point P, point Q and point R and we have to show that these 3 points are co-linear that means these 3 lie in one single line. So what we need to prove is that distance between P and Q plus distance between Q and R should be equal to the distance between the point P and R. If we prove that distance between P and Q plus distance between Q and R is equal to distance between P and R then we can prove that or we can say that P, Q and R are co-linear that means they lie on one line. Also we see the formula that says that if the point P given by x1, y1, z1 and the point Q given by x2, y2, z2 are 2 points then the distance between them P, Q is given by square root of x2 minus x1 the whole square plus y2 minus y1 the whole square plus z2 minus z1 the whole square. So using these two concepts you find out the solution to this question. First of all let the point P be minus 2, 3, 5 let the point Q be 1, 2, 3 and the point R be 7, 0, minus 1. So what we have to show is that these 3 points are co-linear that means we have to show that P, Q plus Q, R should be equal to P, R. Now we find out the distance between P and Q, Q and R and then P and R using this formula that we have just seen. So we see that P, Q that is distance between the point P and Q is given by square root of 1 minus minus 2 is 1 plus 2 so 1 plus 2 the whole square plus 2 minus 3 the whole square plus 3 minus 5 the whole square. This is equal to square root of now 1 plus 2 is 3 so square of 3 is 9 plus 2 minus 3 is minus 1 and square of minus 1 is 1 plus 3 minus 5 is minus 2 and square of minus 2 is 4. So this is equal to square root of 14. Now we find out the distance between Q and R so distance between Q and R is equal to the square root of 7 minus 1 the whole square plus 0 minus 2 the whole square plus minus 1 minus 3 the whole square and this is equal to square root of 7 minus 1 is 6 and square of 6 is 36 plus 0 minus 2 is minus 2 and square of minus 2 is 4 this minus 1 minus 3 is minus 4 and square of minus 4 is 16. So this is equal to square root of 56 and this can be further written as 2 minus 4 is root 14. Now we find out the distance between P and R so P R is given by the square root of 7 minus minus 2 that is 7 plus 2 the whole square plus 0 minus 3 the whole square plus minus 1 minus 5 the whole square. So let's write down what you have just seen 7 plus 2 the whole square plus 0 minus 3 the whole square plus minus 1 minus 5 the whole square this is equal to square root of 7 plus 2 is 9 square of 9 is 81 plus square of minus 3 is 9 plus square of minus 6 is 36 and this is equal to square root of 126 which is equal to 3 root 14. Now we see that P Q plus Q R that is square root of 14 plus twice of square root of 14 is equal to P R that is 3 times square root of 14. So since square root of 14 plus twice of square root of 14 is equal to 3 times square root of 14 therefore P Q plus Q R is equal to P R hence P Q and R that are given by the points minus 2 3 5 1 2 3 and 7 0 minus 1 are collinear. So this is what you are supposed to prove in this question. I hope that you understood the question and enjoyed the session. Have a good day.