What is the distance between the parallel planes 2x+y-3z=13 and 2x+y-3z=35?
Pick any point on the first plane, say,
(0,13,0)
distance from that point to
2x + y - 3z - 35 = 0
= |2(0) + 13 - 3(0) - 35|/√(2^2 + 1^2 + (-3)^2)
= 22/√14 or after rationalizing , 11√14/7
simple. Find a point on the first plane. When x and z are zero, y=13.
Now we find the distance between the point (0,13,0) and the second plane.
d= (2*0+1*13-3*0 -35)/sqrt(4+1+9)
d= 13/sqrt(14)
To find the distance between two parallel planes, we need to find the distance between any point on one plane to the other plane.
In this case, we can use the formula for the distance between a point and a plane. The formula is as follows:
Distance = |ax + by + cz + d| / sqrt(a^2 + b^2 + c^2)
Here, a, b, and c are the coefficients of the variables x, y, and z, respectively, in the equation of the plane. And (x, y, z) represents the coordinates of any point on the plane.
In our case, both planes have the equation 2x + y - 3z. So, the coefficients a, b, and c are all the same. We can choose any point on one of the planes to calculate the distance. For simplicity, let's choose the point (0, 0, 0) as it lies on both planes.
Now, let's calculate the distance:
For the first plane, substitute the coordinates (0, 0, 0) into the equation:
Distance = |2(0) + 0 - 3(0) - 13| / sqrt(2^2 + 1^2 + (-3)^2)
Simplifying, we get:
Distance = |-13| / sqrt(4 + 1 + 9)
Distance = 13 / sqrt(14)
Similarly, for the second plane, substitute the same coordinates (0, 0, 0) into the equation:
Distance = |2(0) + 0 - 3(0) - 35| / sqrt(2^2 + 1^2 + (-3)^2)
Simplifying, we get:
Distance = |-35| / sqrt(4 + 1 + 9)
Distance = 35 / sqrt(14)
Therefore, the distance between the parallel planes 2x + y - 3z = 13 and 2x + y - 3z = 35 is given by the expression 13 / sqrt(14) - 35 / sqrt(14).