Find the distance from the point to the given plane.
(−3,8,7), x−2y−4z=8
To find the distance from a point to a plane, you can use the formula:
distance = |Ax + By + Cz + D| / √(A^2 + B^2 + C^2)
In this formula, (x, y, z) is the coordinate point, and Ax + By + Cz + D = 0 is the equation of the plane.
In your case, the point is P(-3, 8, 7), and the equation of the plane is x - 2y - 4z = 8.
First, let's write the equation of the plane in the form Ax + By + Cz + D = 0:
x - 2y - 4z - 8 = 0
Now, we can identify the values of A, B, C, and D:
A = 1
B = -2
C = -4
D = -8
Next, substitute the coordinates of the point (-3, 8, 7) and the values of A, B, C, and D into the formula:
distance = |1 * (-3) + (-2) * 8 + (-4) * 7 + (-8)| / √(1^2 + (-2)^2 + (-4)^2)
Simplifying further:
distance = |-3 - 16 - 28 - 8| / √(1 + 4 + 16)
distance = |-55| / √21
Finally, we calculate the distance:
distance = 55 / √21
Therefore, the distance from the point (-3, 8, 7) to the plane x - 2y - 4z = 8 is 55 / √21 (approximately 11.14).