How can I find the equation of a line perpendicular to a plane in 3D?

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To find the equation of a line perpendicular to a plane in 3D that passes through point A(x1, y1, z1), one must first determine the normal vector of the plane defined by points A, B(x2, y2, z2), and C(x3, y3, z3). This can be achieved by calculating the cross product of the vectors AB and AC, which will yield the normal vector. The line can then be expressed in Cartesian form, using point A and the direction of the normal vector. It is essential that the line is parallel to the normal vector of the plane to maintain perpendicularity. Understanding these concepts is crucial for solving related geometric problems in three-dimensional space.
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Homework Statement


How would I find the equation of a line which passes through A(x1, y1, z1) and is perpendicular to A(x1, y1, z1) B(x2, y2, z2), C(x3, y3, z3)

Homework Equations


Cartesian form of a line in \mathbb{R}^3

The Attempt at a Solution


Not sure if this is right but would finding the equation of the plane which contains all three points and then finding the equation of the line that passes through A and that plane be correct?
 
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Yes, but your line needs to be [as you say] perpendicular to the plane, which is equivalent to being parallel to the normal to the plane.

If you don't really care about the plane, you have enough information for find a normal-vector to the plane... by using cross-products.
 

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