Question

Use the dot product to find the distance ( d) from the point A(2,-6,1) to the line ( l ) through B(3,4,-2) and C(7,-1,5).

Answer 1

To find the distance from a point to a line in three-dimensional space using the dot product, calculate the directional vector of the line, the vector from one point on the line to the given point, and then project this vector onto the line's direction vector to find the perpendicular distance.

Step-by-step explanation:

To find the distance (d) from point A(2, -6, 1) to the line through points B(3, 4, -2) and C(7, -1, 5), we can use the dot product and a vector projection method. The process involves several steps:

1. Find the vector representing the direction of line l, which is obtained by subtracting the coordinates of point B from those of point C, providing the directional vector ℓ = C - B.
2. Calculate the vector from point B to point A as vector BA.
3. Compute the projection of vector BA onto vector ℓ to find the component of BA that is parallel to ℓ.
4. Use the projection to determine the perpendicular distance from point A to line l, which is the magnitude of the vector BA minus its projection onto ℓ.

In our case, the directional vector ℓ is (7 - 3, -1 - 4, 5 - (-2)) or (4, -5, 7). Vector BA is (2 - 3, -6 - 4, 1 - (-2)) or (-1, -10, 3). The dot product of vectors BA and ℓ gives us the length of the projection of BA onto ℓ. Finally, the distance d can be calculated using the magnitude of BA and the magnitude of its projection onto ℓ.