 Hi friends, I am Purva and today we will discuss the following question. Show that the points A with coordinates 1, minus 2, minus 8, B with coordinates 5, 0, minus 2 and C with coordinates 11, 3, 7 are collinear and find the ratio in which B divides AC. Let us now begin with the solution. Now we are given A, B and C, so let us find vector AB first. So we have vector AB is equal to 5i cap plus 0j cap minus 2k cap minus 1i cap minus 2j cap minus 8k cap. This is equal to 5 minus 1i cap plus 0 plus 2j cap plus minus 2 plus 8k cap. And we get this is equal to 4i cap plus 2j cap plus 6k cap. So we have got vector AB is equal to 4i cap plus 2j cap plus 6k cap. Now this implies vector AB is equal to now taking out two common from right hand side we get 2 into 2i cap plus j cap plus 3k cap. And we mark this as equation 1. Now we will find vector BC. So vector BC is equal to 11i cap plus 3j cap plus 7k cap minus 5i cap plus 0j cap minus 2k cap. And this is equal to 11 minus 5i cap plus 3 minus 0j cap plus 7 plus 2k cap. This is further equal to 6i cap plus 3j cap plus 9k cap. So we get vector BC is equal to now taking out three common from right hand side we get 3 into 2i cap plus j cap plus 3k cap. Now this is equal to 3 into now from equation 1 we can clearly see that vector AB is equal to 2 into 2i cap plus j cap plus 3k cap. So we get 2i cap plus j cap plus 3k cap is equal to vector AB upon 2. So we get here vector AB upon 2 and this is from equation 1. So we get vector BC is equal to 3 upon 2 into vector AB. Or we can write this as vector AB is equal to 2 upon 3 into vector BC. Now we know that if vector A is equal to lambda vector B where lambda is any scalar then from this we can say that either vector A is parallel to vector B or vector A and vector B are collinear. Now here vector AB is equal to 2 by 3 into vector BC. So we can say either vector BC is parallel to vector AB or they are collinear. But point B is common in vector AB and vector BC. Hence we have vector BC and vector AB are collinear. And B divides AC in the ratio 2 is to 3. Hence we write our answer as 2 is to 3. Hope you have understood the solution. Bye and take care.