 Good morning friends. I am Purva and today we will discuss the following question. Find the angle between the planes whose vector equations are vector r dot 2i cap plus 2j cap minus 3k cap is equal to 5 and vector r dot 3i cap minus 3j cap plus 5k cap is equal to 3. Now let there be two planes with equations. Vector r dot vector n1 is equal to d1 and vector r dot vector n2 is equal to d2. Then the angle theta between the planes is the angle between normal of the planes and here we know that vector n1 and vector n2 are the two normals of the planes. So the angle theta between the planes is given by cos theta is equal to mod of vector n1 dot vector n2 upon mod of vector n1 into mod of vector n2. So this is the key idea behind our question. Let us begin with the solution now. Now we have given the equations of the planes as vector r dot 2i cap plus 2j cap minus 3k cap is equal to 5 and vector r dot 3i cap minus 3j cap plus 5k cap is equal to 3. Now comparing this equation with this equation of the key idea we can clearly see that here the normal that is vector n1 is equal to 2i cap plus 2j cap minus 3k cap and comparing this equation with the second equation in the key idea we can clearly see that here normal that is vector n2 is equal to 3i cap minus 3j cap plus 5k cap. So we write here vector n1 is equal to 2i cap plus 2j cap minus 3k cap and vector n2 is equal to 3i cap minus 3j cap plus 5k cap. Now by key idea we also know that the angle theta between the planes is given by cos theta is equal to mod of vector n1 dot vector n2 upon mod of vector n1 into mod of vector n2. Now vector n1 dot vector n2 is equal to vector n1 is equal to 2i cap plus 2j cap minus 3k cap dot vector n2 is equal to 3i cap minus 3j cap plus 5k cap and this is equal to 2 into 3 is 6 2 into minus 3 is minus 6 minus 3 into 5 is minus 15. So we get this is equal to minus 15. Now mod of vector n1 is equal to under root of 2 square plus 2 square plus minus 3 whole square because we have vector n1 is equal to 2i cap plus 2j cap minus 3k cap and we get this is equal to under root of 4 plus 4 plus 9 and this is equal to under root 17. Also mod of vector n2 is equal to under root of 3 square plus minus 3 whole square plus 5 square and this is equal to under root of 9 plus 9 plus 25 and this is equal to under root 43. So mod of vector n1 into mod of vector n2 is equal to under root 17 into under root 43 which is equal to under root 731. Now putting all these values in this equation 1 we get cos theta is equal to mod of now vector n1 dot vector n2 is equal to minus 15. So we have minus 15 upon mod of vector n1 into mod of vector n2 is equal to under root 731. So we get under root 731 this is equal to 15 upon under root 731. So we have got cos theta is equal to 15 upon under root 731 and this implies theta is equal to cos inverse 15 upon under root 731. So we have got our answer as cos inverse 15 upon under root 731. Hope you have understood the solution. Bye and take care.