Question

The distance d of a point P to the line through points A and B is the length of the component of AP¯¯¯¯¯¯¯¯ that is orthogonal to AB¯¯¯¯¯¯¯¯, as indicated in the diagram. So the distance from P=(−3,3,5) to the line through the points A=(−4,4,5) and B=(5,1,−4) is

Answer 1

The distance from point P=(-3,3,5) to the line through points A=(-4,4,5) and B=(5,1,-4) is 4.24 units.

Step-by-step explanation:

The distance between a point and a line can be calculated by finding the projection of the vector AP onto the vector AB, dividing it by the magnitude of AB.

In this case, the vector AP=(-3+4, 3-4, 5-5)=(1, -1, 0), and AB=(5+4, 1-4, -4-5)=(9, -3, -9). The dot product of AP and AB is 19 + (-1)(-3) + 0*(-9) = 12. The magnitude of AB is sqrt(9^2 + (-3)^2 + (-9)^2) = sqrt(81+9+81) = sqrt(171) = 13.07.

Thus, the projection of AP onto AB is 12 / 13.07 = 0.92. Therefore, the distance is 0.92 * magnitude of AB = 0.92 * 13.07 = 12 units.

Answer 2

The distance from point P=(-3,3,5) to the line through points A=(-4,4,5) and B=(5,1,-4) is approximately 0.92 units.

Step-by-step explanation:

To find the distance from point P=(-3,3,5) to the line through points A=(-4,4,5) and B=(5,1,-4), we can follow these steps:

1. Find the vector AB by subtracting the coordinates of point A from the coordinates of point B.

AB = (5-(-4), 1-4, -4-5) = (9, -3, -9)

2. Find the vector AP by subtracting the coordinates of point A from the coordinates of point P.

AP = (-3-(-4), 3-4, 5-5) = (1, -1, 0)

3. Calculate the dot product of AP and AB. This can be done by multiplying the corresponding components of the two vectors and summing the results.

AP · AB = (1 * 9) + (-1 * -3) + (0 * -9) = 9 + 3 + 0 = 12

4. Calculate the magnitude of vector AB using the formula ||AB|| =√(x² + y² + z² ), where x, y, and z are the components of AB.

||AB|| = √(9² + (-3)² + (-9)² ) = √(81 + 9 + 81) = √(171) ≈ 13.08

5. Divide the dot product (AP · AB) by the magnitude of AB (||AB||).

Distance = |AP · AB| / ||AB|| = |12| / 13.08 = 12 / 13.08 ≈ 0.92

Therefore, the distance from point P=(-3,3,5) to the line through points A=(-4,4,5) and B=(5,1,-4) is approximately 0.92 units.