dtl42 said:I tried finding the vector perpendicular to the plane, [tex]<3,2,-1>[/tex], but I'm not sure what to do with it.
When we say "Find Line in Plane Passing Through (2,3,6)", we are referring to finding the equation of a straight line that lies on a specific plane and passes through the point (2,3,6). This means that all the points on the line will also lie on the given plane and must pass through the point (2,3,6).
To find the equation of a line passing through a specific point, we need to know the coordinates of that point and the slope of the line. With this information, we can use the point-slope form of a line equation, y - y1 = m(x - x1), where (x1, y1) are the coordinates of the given point and m is the slope. We can also use other forms of the line equation, such as slope-intercept form or two-point form, depending on the given information.
To determine a plane passing through a given point, we need to know the coordinates of that point and the normal vector to the plane. The normal vector is a vector that is perpendicular to the plane. With this information, we can use the point-normal form of a plane equation, a(x - x1) + b(y - y1) + c(z - z1) = 0, where (x1, y1, z1) are the coordinates of the given point and a, b, and c are the components of the normal vector.
Yes, there can be infinite lines passing through a given point and lying on the same plane. This is because a plane is a two-dimensional surface and a line is a one-dimensional object. As long as the line lies on the plane and passes through the given point, it can be considered a solution.
If the line lies entirely on the plane, then it is parallel to the plane. If the line is perpendicular to the normal vector of the plane, then it is perpendicular to the plane. In other words, the dot product between the line's direction vector and the normal vector of the plane is 0 for perpendicular lines, and it is non-zero for parallel lines.