determine the angle between the planes 2x + 2y + 7z = 8 and 3x - 4y + 4z = 5
let Ø be the angle between their normals
(2,2,7)∘(3,-4,4) = √57√41cosØ
cosØ = (6 - 8 + 28)/(√57√41)
Ø = appr. 57.5°
so the two planes form an angle of 180-57.5 or 122.5°
To determine the angle between two planes, we need to find the normal vectors of the planes first. The normal vector of a plane is a vector perpendicular to the plane.
The equation of the first plane is 2x + 2y + 7z = 8. To find the normal vector of this plane, we can rewrite the equation in the form Ax + By + Cz = D, where A, B, C, and D are coefficients:
2x + 2y + 7z - 8 = 0
Comparing this equation with the general form Ax + By + Cz = 0, we can see that the coefficients of x, y, and z are 2, 2, and 7, respectively. Therefore, the normal vector of the first plane is (2, 2, 7).
Similarly, for the second plane with the equation 3x - 4y + 4z = 5, we can rewrite it as:
3x - 4y + 4z - 5 = 0
Comparing coefficients, we find that the normal vector of the second plane is (3, -4, 4).
Now that we have the normal vectors of both planes, we can determine the angle between them using the dot product. The dot product of two vectors, A and B, is given by A · B = |A| |B| cos(theta), where |A| and |B| are the magnitudes of vectors A and B, and theta is the angle between them.
Let's calculate the dot product of the two normal vectors:
(2, 2, 7) · (3, -4, 4) = (2 * 3) + (2 * -4) + (7 * 4) = 6 - 8 + 28 = 26
The magnitudes of the normal vectors can be calculated using the formula |A| = sqrt(Ax^2 + Ay^2 + Az^2). For the first normal vector:
|A| = sqrt(2^2 + 2^2 + 7^2) = sqrt(4 + 4 + 49) = sqrt(57)
And for the second normal vector:
|B| = sqrt(3^2 + (-4)^2 + 4^2) = sqrt(9 + 16 + 16) = sqrt(41)
Now, let's substitute the values into the dot product formula:
26 = sqrt(57) * sqrt(41) * cos(theta)
To find cos(theta), we can rearrange the equation:
cos(theta) = 26 / (sqrt(57) * sqrt(41))
Finally, we can find the angle theta by taking the inverse cosine (arccos) of cos(theta):
theta = arccos(26 / (sqrt(57) * sqrt(41)))
Using a calculator or software, we can find the value of theta, which would be the angle between the two planes.