Final answer:
To find the cosine of the angle between the planes x + y + z = 0 and 2x + 3y + z = 0, we need to determine the normal vectors of these planes and use the dot product formula to find the cosine of the angle.
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 determine the normal vectors of these planes.
The normal vector of the plane x + y + z = 0 is (1, 1, 1) and the normal vector of the plane 2x + 3y + z = 0 is (2, 3, 1).
The cosine of the angle between two vectors can be calculated using the dot product formula: cosθ = (A·B) / (|A||B|).
Using the dot product formula, we can calculate the cosine of the angle between the normal vectors as follows:
cosθ = ((1)(2) + (1)(3) + (1)(1)) / (√(1² + 1² + 1²) * √(2² + 3² + 1²))
cosθ = (2 + 3 + 1) / (√3 * √14)
cosθ = 6 / (√42)
Thus, the cosine of the angle between the planes x + y + z = 0 and 2x + 3y + z = 0 is 6 / (√42).