Final answer:
To find the angle between two planes, we need to find the direction vectors of the planes and use the dot product formula. To find the line of intersection, we can set one variable as a parameter, substitute it back into the equations, and solve for the remaining variables.
Step-by-step explanation:
To find the angle between two planes, we need to find the direction vectors of the planes and use the dot product formula. The direction vectors of the planes can be found by taking the coefficients of x, y, and z in their respective equations. The dot product of the direction vectors will give us the cosine of the angle between the planes. To find the line of intersection, we can set one variable (say, z) as a parameter, substitute it back into the equations, and solve for the remaining variables (x and y).
(a) For the given planes, the direction vectors are [-4, 1, 1] and [20, -5, 4]. Using the dot product formula, we have cos(theta) = (-4)(20) + (1)(-5) + (1)(4) / ((sqrt((-4)^2 + 1^2 + 1^2))(sqrt((20)^2 + (-5)^2 + 4^2))). Solving this expression will give us the angle between the two planes.
(b) To find the line of intersection, let's assume that z is the parameter. We can choose one equation and solve for x and y in terms of z. Substituting the value of z back into the other equation will give us the parametric equations for the line of intersection.