 Good morning friends I am Pooh 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 planes are 7x plus 5y plus 6z plus 30 is equal to 0 and 3x minus y minus 10z plus 4 is equal to 0. Now let the equations of the two 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 planes as 7x plus 5y plus 6z plus 30 is equal to 0 and 3x minus y minus 10z plus 4 is equal to 0. Now in the key idea we have equations of the planes as vector r dot vector n1 is equal to d1 and vector r dot vector n2 is equal to d2. Now suppose the position vector r is equal to xi cap plus yj cap plus zk cap and normal vector n1 is equal to ai cap plus bj cap plus ck cap. Then by key idea we know that vector r dot vector n1 is equal to d1. Now this 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 and this is equal to d1. This further 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. This is the Cartesian equation of the plane. Now comparing this equation with this equation we can clearly see that here a is equal to 7, b is equal to 5, c is equal to 6 so we get the normal vector n1 as 7i cap plus 5j cap plus 6k cap. Again comparing this equation with this equation we can see that here a is equal to 3, b is equal to minus 1 and c is equal to minus 10 so we get the normal vector n2 as 3i cap minus j cap minus 10k cap so we get the normal vectors of the planes as vector n1 is equal to 7i cap plus 5j cap plus 6k cap and vector n2 is equal to 3i cap minus j cap minus 10k cap. Now the direction ratios of vector n1 are 75 and 6 and the direction ratios of vector n2 are 3 minus 1 minus 10 taking the proportions we get 7 upon 3 5 upon minus 1 and 6 upon minus 10. Now here we can clearly see that 7 upon 3 is not equal to 5 upon minus 1 which is not equal to 6 upon minus 10 so we get vector n1 is not parallel to vector n2 and thus the planes are not parallel to each other. Now we will see that whether the planes are perpendicular or not so for that first we will find vector n1 dot vector n2 and this is equal to vector n1 is equal to 7i cap plus 5j cap plus 6k cap dot vector n2 is equal to 3i cap minus j cap minus 10k cap this is equal to 7 into 3 is 21 5 into minus 1 is minus 5 and 6 into minus 10 is minus 60 and we get this is equal to minus 44. So we have got vector n1 dot vector n2 is equal to minus 44 that is vector n1 dot vector n2 is not equal to 0. Now by key idea we know that if planes are perpendicular then we have vector n1 dot vector n2 is equal to 0 so here we get thus planes are not perpendicular. In the question we are given that determine whether the given planes are parallel or perpendicular and in case they are neither find the angle between them. So here we have seen that the planes are neither parallel nor perpendicular so we will find the angle between them. Again by key idea we 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. So here we have 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 we have mod of vector n1 is equal to under root of 7 square plus 5 square plus 6 square because we have vector n1 is equal to 7i cap plus 5j cap plus 6k cap. This is equal to under root of 49 plus 25 plus 36 which is equal to under root of 110. Now mod of vector n2 is equal to under root of 3 square plus minus 1 whole square plus minus 10 whole square and we get this is equal to under root of 9 plus 1 plus 100. This is equal to under root of 110. So we get mod of vector n1 into mod of vector n2 is equal to under root of 110 into under root of 110 and we get this is equal to 110. Now we mark this as equation 1, we mark this as 2 and we mark this as 3. So putting 1 and 3 into we get cos theta is equal to mod of minus 44 upon 110 and we get this is equal to 44 upon 110. Translating the common factors here we get 2 in numerator and 5 in denominator. So we get this implies cos theta is equal to 2 upon 5 which further implies theta is equal to cos inverse 2 upon 5. Thus we get our answer as cos inverse 2 upon 5. Hope you have understood the solution. Bye and take care.