What is the relationship of the given plane 2x-10y-2z+2=0 to each of the planes below? Is it coincident, parallel or intersecting? 2. -3x-4y+4z =-33

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to check if the given planes intersect each other, hence, you need to evaluate the cross product of normal vectors to the planes, such that:

`bar (n_1) = <2,-10,-2>`

`bar (n_1) = <-3,-4,4>`

`bar (n_1) X bar (n_2) = [(bar i, bar j, bar k),(2, -10, -2),(-3, -4, 4)]`

`bar (n_1) X bar (n_2) = -40 bar i - 8 bar k + 6 bar j - 30 bar k - 8 bar i - 8 bar j`

`bar (n_1) X bar (n_2) = -48 bar i - 2 bar j - 38 bar k`

Since the result of cross product is the vector `bar v = -48 bar i - 2 bar j - 38 bar k` yields that the plane `2x-10y-2z+2=0` intersects the plane `-3x-4y+4z =-33` at the line whose directional vector is `bar (n_1) X bar (n_2) = bar v = -48 bar i - 2 bar j - 38 bar k.`