For the given point and line, find by projection the point on the line that is closest to the given point and use perp to find the distance from the point to the line.
P (3,-5)
line: x = (2,-4)+t(3,-4)
To find the point on the line that is closest to the given point, we can use projection. Let's follow these steps:
1. Determine the direction vector of the line. Given that the line is parameterized as x = (2,-4) + t(3,-4), the direction vector is (3,-4).
2. Find a vector from a point on the line to the given point. We can choose any point on the line, so let's use the point (2,-4) as our reference point. The vector from (2,-4) to (3,-5) is (3,-5) - (2,-4) = (1,-1).
3. Find the projection of the vector from step 2 onto the direction vector from step 1. To calculate the projection, multiply the vector from step 2 by the normalized direction vector (3,-4) / ||(3,-4)||, where ||(3,-4)|| represents the magnitude of the vector (3,-4).
The magnitude of (3,-4) can be calculated as follows:
||(3,-4)|| = sqrt((3^2) + (-4^2)) = sqrt(9 + 16) = sqrt(25) = 5
Normalized direction vector:
(3,-4) / ||(3,-4)|| = (3,-4) / 5 = (3/5, -4/5)
Projection: (1,-1) dot (3/5, -4/5) * (3/5, -4/5) = ((1 * 3/5) + (-1 * -4/5))(3/5, -4/5) = (7/5, 12/5)
4. Calculate the point on the line that is closest to the given point. To find this point, add the projection vector from step 3 to the reference point (2,-4).
Closest point on the line: (2,-4) + (7/5, 12/5) = (2 + 7/5 , -4 + 12/5) = (17/5, -8/5)
Now let's move on to finding the distance from the given point (3,-5) to the line using the perpendicular method.
5. Determine a vector from the closest point on the line (17/5, -8/5) to the given point (3,-5). The vector from (17/5, -8/5) to (3,-5) is (3,-5) - (17/5, -8/5) = (3 - 17/5, -5 + 8/5) = (-2/5, -7/5).
6. Calculate the distance from the given point to the line by taking the magnitude of the vector from step 5.
Distance: ||(-2/5, -7/5)|| = sqrt((-2/5)^2 + (-7/5)^2) = sqrt(4/25 + 49/25) = sqrt(53/25) = sqrt(53) / 5
Therefore, the point on the line that is closest to the given point (3,-5) is (17/5, -8/5), and the distance from the point to the line is sqrt(53) / 5.