Question

Find the cosine of the angle between the planes x + y + z = 0 and 2x + 3y + z = 0.

Answer 1

To find the cosine of the angle between the planes x + y + z = 0 and 2x + 3y + z = 0, we need to find the normal vectors of both planes and then find the dot product of the normal vectors.

Step-by-step explanation:

To find the cosine of the angle between the planes x + y + z = 0 and 2x + 3y + z = 0, we need to find the normal vectors of both planes and then find the dot product of the normal vectors. The cosine of the angle between two vectors is equal to the dot product of the vectors divided by the product of their magnitudes.

The normal vectors of the planes are (1, 1, 1) and (2, 3, 1), respectively.

The dot product is (1)(2) + (1)(3) + (1)(1) = 6 + 3 + 1 = 10.

The magnitudes of the vectors are
`√(1^2 + 1^2 + 1^2)` =
`√(3)` and
`√(2^2 + 3^2 + 1^2)` =
`√(14)`.

Therefore, the cosine of the angle between the planes is
`10 / √(3) *√(14)`.