Final answer:
To find the angle of intersection between two planes, calculate the angle between their normal vectors using the dot product and magnitudes, then find the inverse cosine of that value to get the angle in radians and multiply by 180/π to convert to degrees.
Step-by-step explanation:
To find the angle of intersection between two planes, you need to find the angle between their normal vectors. The normal vector for the first plane is N1 = (-4, 2, 5), and for the second plane N2 = (-2, -5, -3). The angle θ between the two normal vectors can be found using the dot product formula and the cosine:
N1 · N2 = |N1||N2|cos(θ)
We first compute the dot product:
N1 · N2 = (-4)×(-2) + (2)×(-5) + (5)×(-3) = 8 - 10 - 15 = -17
Then, compute the magnitudes of N1 and N2:
|N1| = √((-4)² + 2² + 5²) = √(16 + 4 + 25) = √45
|N2| = √((-2)² + (-5)² + (-3)²) = √(4 + 25 + 9) = √38
Now we can find the cosine of the angle:
cos(θ) = (N1 · N2) / (|N1||N2|) = -17 / (√45 * √38)
Finally, we take the inverse cosine to get the angle θ:
θ = cos⁻¹(-17 / (√45 * √38)) in radians
Then, convert θ to degrees:
θ in degrees = θ in radians × (180/π)
Make sure to calculate this using a calculator to get the angle θ in radians and degrees.