Final answer:
To find an equation of a plane through a point that is orthogonal to a line, we need to use the dot product. We can find a vector perpendicular to the line by taking the cross product of the line's direction vector with any other vector. The equation of the plane will be n · (r - P) = 0.
Step-by-step explanation:
To find an equation of a plane through a point that is orthogonal to a line, we need to use the dot product. Let's say the point is P and the line is given by the vector equation r = a + tb.
We can find a vector perpendicular to the line by taking the cross product of the line's direction vector b with any other vector. Let's call this vector n. Then, the equation of the plane will be n · (r - P) = 0.
For example, if the line is given by r = (1, 2, -1) + t(2, -3, 4) and the point is P(3, -1, 2), we can find n = (2, -3, 4) × (1, 2, -1) = (-13, -6, -1). The equation of the plane would then be -13(x - 3) - 6(y + 1) - 1(z - 2) = 0.