Final answer:
The normal vectors ℓ₁ and ℓ₂ for the given planes are (8, 32, -24) and (-24, 48, 56) respectively. By calculating their dot product and checking the conditions for parallelism or perpendicularity, we can determine the relationship between the planes and find the angle between them if needed.
Step-by-step explanation:
To find the normal vectors for the given planes, we look at the coefficients of x, y, and z in the plane equations. The normal vector ℓ₁ for the plane 8x + 32y - 24z = 1 is (8, 32, -24). Similarly, the normal vector ℓ₂ for the plane -24x + 48y + 56z = 0 is (-24, 48, 56).
The dot product ℓ₁.ℓ₂ is calculated by multiplying corresponding components of the two vectors and then adding them up:
ℓ₁.ℓ₂ = (8)(-24) + (32)(48) + (-24)(56).
The planes are considered parallel if their normal vectors are scalar multiples of each other, perpendicular if their dot product equals zero, and neither if neither condition is met. To find the angle between the planes when they are neither parallel nor perpendicular, we use the formula:
cos(θ) = ℓ₁.ℓ₂ / (|ℓ₁||ℓ₂|),
and then calculate θ by taking the inverse cosine.