Find the perpendicular distance of the point A(1,0,1) to the line through the points B(2,3,4) and C(-1,1,-2)

To find the perpendicular distance from the point A(1,0,1) to the line defined by points B(2,3,4) and C(-1,1,-2), we can follow these steps:

Step 1: Determine the Direction Vector of the Line

The direction vector of the line can be found by subtracting the coordinates of point B from point C:

• Direction vector, D = C - B = (-1 - 2, 1 - 3, -2 - 4) = (-3, -2, -6)

Step 2: Find the Vector from Point B to Point A

Next, we calculate the vector from point B to point A:

• Vector BA = A - B = (1 - 2, 0 - 3, 1 - 4) = (-1, -3, -3)

Step 3: Calculate the Cross Product

The cross product of vectors BA and D gives us a vector perpendicular to both:

• BA × D = |i j k|
• |-1 -3 -3|
• |-3 -2 -6|

Calculating the determinant:

• i: (-3)(-6) - (-3)(-2) = 18 - 6 = 12
• j: -((-1)(-6) - (-3)(-3)) = - (6 - 9) = 3
• k: (-1)(-2) - (-3)(-3) = 2 - 9 = -7

The cross product is (12, 3, -7).

Step 4: Calculate the Magnitude of the Cross Product

The magnitude of the cross product vector is:

• |BA × D| = √(12² + 3² + (-7)²) = √(144 + 9 + 49) = √202

Step 5: Calculate the Magnitude of the Direction Vector

The magnitude of the direction vector D is:

• |D| = √((-3)² + (-2)² + (-6)²) = √(9 + 4 + 36) = √49 = 7

Step 6: Find the Perpendicular Distance

The perpendicular distance d from point A to the line is given by:

d = |BA × D| / |D| = √202 / 7

Thus, the perpendicular distance from point A(1,0,1) to the line through points B(2,3,4) and C(-1,1,-2) is √202 / 7.