 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, find the angle between them. And the equation of the planes are 2x-y plus 3z-1 is equal to 0 and 2x-y plus 3z plus 3 is equal to 0. Now let equations of 2 planes be 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 the normal of the planes and 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 we have vector n1 is parallel to vector n2 and if the planes are perpendicular then we have 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 equations of the planes as 2x-y plus 3z-1 is equal to 0 and 2x-y plus 3z plus 3 is equal to 0. Now in the key idea we are given the vector equations of the planes as vector r dot vector n1 is equal to d1 and vector r dot vector n2 is equal to d2. We consider the first equation that is vector r dot vector n1 is equal to d1. Now here if the position vector r is equal to xi cap plus yj cap plus zk cap and the normal vector n1 is equal to ai cap plus bj cap plus ck cap then we have vector r dot vector n1 is equal to d1 implies vector r is equal to xi cap plus yj cap plus zk cap dot vector n1 is equal to ai cap plus bj cap plus ck cap is equal to d1. Now this implies ax plus by plus cz is equal to d1 or we can write this as ax plus by plus cz minus d1 is equal to 0. Now this is the Cartesian equation of the plane comparing this equation with this equation we can clearly see that here a is equal to 2, b is equal to minus 1 and c is equal to 3. So we get the normal vector of this plane as 3i cap minus j cap plus 3k cap. Now on comparing this equation with this equation we see that here a is equal to 2, b is equal to minus 1 and c is equal to 3. So we get the normal vector of this plane as 2i cap minus j cap plus 3k cap. So the normal vectors of the planes are vector n1 is equal to 2i cap minus j cap plus 3k cap and vector n2 is equal to 2i cap minus j cap plus 3k cap. Now we see that the direction ratios of vector n1 are 2 minus 1, 3 and the direction ratios of vector n2 are 2 minus 1, 3 taking the proportions we get 2 upon 2 comma minus 1 upon minus 1 comma 3 upon 3 or we get 1 comma 1 comma 1. Now 1 is equal to 1 which is equal to 1. It means vector n1 is parallel to vector n2 thus the planes are parallel to each other because by the key idea we know that if the planes are parallel then we have vector n1 is parallel to vector n2. So we write our answer as the planes are parallel. Hope you have understood the solution. Bye and take care.