 Hi friends, I am Purva and today we will discuss the following question. In the following, determine whether the given planes are parallel or perpendicular, and in case they are neither, then find the angle between them. And the planes are 2x minus 2y plus 4z plus 5 is equal to 0 and 3x minus 3y plus 6z minus 1 is equal to 0. Suppose there are 2 planes with equations vector r dot vector n1 is equal to d1 and vector r dot vector n2 is equal to d2. Now here vector n1 and vector n2 are normals to the plane. Then the angle theta between the planes is the angle between normal of the planes which 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 if the planes are parallel, then vector n1 is parallel to vector n2 and if the planes are perpendicular, then vector n1 dot vector n2 is equal to 0. So this is the key idea behind our question. Let us begin with the solution now. Now we are given the equation of the planes are 2x minus 2y plus 4z plus 5 is equal to 0 and 3x minus 3y plus 6z minus 1 is equal to 0. From the equation of the planes we get the normal vectors as vector n1 is equal to 2i cap minus 2j cap plus 4k cap and vector n2 is equal to 3i cap minus 3j cap plus 6k cap. Now the direction ratios of vector n1 are 2 comma minus 2 comma 4 and the direction ratios of vector n2 are 3 comma minus 3 comma 6 taking the proportions we get 2 upon 3 comma minus 2 upon minus 3 comma 4 upon 6 or we can write this as 2 upon 3 comma cancelling out minus sign we get 2 upon 3 comma cancelling out common factor 2 here from numerator and denominator we get 2 upon 3. So we have 2 upon 3 is equal to 2 upon 3 is equal to 2 upon 3. This means the direction ratios are proportionate hence the planes are parallel to each other. Thus we have got our answer as the planes are parallel. Hope you have understood the solution. Bye and take care.