Let A(α, 4, 7) and B(3, β, 8) be two points in space. If the YZ-plane and ZX-plane divide the line segment joining the points A and B in the ratio 2:3 and 4:5 respectively, then the point C which divides AB externally in the ratio α : β is:

If θ is the acute angle between the planes x + 2y + 2z = 0 and 2x - 2y + z = 0, then cos(θ) is equal to:

Find the vector equation for the line passing through the points A(2, 1, 5) and B(4, 5, 7).

Find the angle between the pair of lines given by: r = î - 2ȷ - 4k̂ + λ(-2î + 2ȷ + k̂) and r = 4î + ȷ - 3k̂ + μ(2î - 4ȷ - 3k̂).