 Hi friends, I am Purva and today we will discuss the following question. Show that the points a, b and c with position vectors, vector a is equal to 3i cap minus 4j cap minus 4k cap, vector b is equal to 2i cap minus j cap plus k cap and vector c is equal to i cap minus 3j cap minus 5k cap respectively from the vertices of a right angle triangle. Let us now begin with the solution. Now let a is given by 3i cap minus 4j cap minus 4k cap, b is given by 2i cap minus j cap plus k cap and c is given by i cap minus 3j cap minus 5k cap. Then we have vector a, b is equal to 2i cap minus j cap plus k cap minus 3i cap minus 4j cap minus 4k cap. This is equal to 2 minus 3i cap plus minus 1 plus 4j cap plus 1 plus 4k cap and we have this is equal to minus 1i cap plus 3j cap plus 5k cap. Now vector bc is given by i cap minus 3j cap minus 5k cap minus 2i cap minus j cap plus k cap and we have this is equal to 1 minus 2i cap plus minus 3 plus 1j cap plus minus 5 minus 1k cap and this is equal to minus 1i cap minus 2j cap minus 6k cap. Finally vector ca is equal to 3i cap minus 4j cap minus 4k cap minus i cap minus 3j cap minus 5k cap and this is equal to 3 minus 1i cap plus minus 4 plus 3j cap plus minus 4 plus 5k cap and this is equal to 2i cap minus j cap plus k cap. Now mod of vector ab is given by under root of minus 1 square plus 3 square plus 5 square and this is equal to under root of 1 plus 9 plus 25 which is equal to root 35. So we have got mod of vector ab is equal to root 35 and we mark this as equation 1. Mod of vector bc is equal to under root of minus 1 square plus minus 2 square plus minus 6 square and this is equal to under root of 1 plus 4 plus 36 and we get this is equal to under root 41. So we have got mod of vector bc is equal to root 41 and we mark this as equation 2. Finally mod of vector ca is equal to under root of 2 square plus minus 1 square plus 1 square and we have this is equal to under root of 4 plus 1 plus 1 which is equal to under root 6. So we have got mod of vector ca is equal to root 6 and we mark this as equation 3. Now to show that the points ab and c form the vertices of a right angle triangle, we will show that the sum of squares of magnitude of any of the two vectors given in 1, 2 and 3 is equal to the square of magnitude of the third vector. Now from 1, 2 and 3 we can clearly see that mod of vector bc square is equal to 41 and we have this is equal to 35 plus 6 we can write 41 as 35 plus 6 and this is equal to now from 1 we can see that 35 is equal to mod of vector ab square. So we have mod of vector ab square plus now from 3 we can see that 6 is equal to mod of vector ca square. So we have mod of vector ca square. So we have seen mod of vector bc square is equal to mod of vector ab square plus mod of vector ca square. Hence we have the triangle is the right angle triangle. This is our answer. Hope you have understood the solution. Bye and take care.