dodgedanpei said:Homework Statement
The plane that passes through the point (1, 6, 4) and contains the line
x = 1 + 2t; y = 2 - 3t; z = 3 - t where t is an element of R
Homework Equations
x = 1 + 2t; y = 2 - 3t; z = 3 - t
The Attempt at a Solution
Let L be the solution.
L = (1,6,4) - ?
t = (x -1)/ 2 = (2-y)/3 = 3-z
This doesn't make any sense at all. First off, <1, 2, 3> is a vector from the origin to the point (1, 2, 3) on the line. Second, the vector t<2, -3, -1> = <2t, -3t, -t> is a vector that has the same direction as the line.dodgedanpei said:Well I tried making 2 vectors by using the 3 equations.
I got
vector x = t(2,-3,-1) = (1,2,3)
dodgedanpei said:But the two vectors are meant to be s(0,-4,7) + t(-8,0,25) , where s and t are real numbers.
A plane in mathematics is a two-dimensional flat surface that extends infinitely in all directions. It is defined by three non-collinear points or by a point and two non-parallel vectors.
To find the equation of a plane using three points, you can use the formula ax + by + cz + d = 0, where a, b, and c are the coefficients of the variables x, y, and z, and d is a constant. Substitute the coordinates of the three points into this formula and solve for a, b, c, and d.
Yes, you can find the equation of a plane with only one point and a normal vector. The equation is given by (x-x0, y-y0, z-z0) · n = 0, where (x0, y0, z0) is the given point and n is the normal vector.
The cartesian form of a plane's equation is ax + by + cz + d = 0, where a, b, and c are the coefficients of the variables x, y, and z, and d is a constant. This form is useful for finding the distance between a point and the plane. The vector form of a plane's equation is (x,y,z) = (x0, y0, z0) + s(a,b,c) + t(d,e,f), where (x0, y0, z0) is a point on the plane, (a,b,c) and (d,e,f) are two non-parallel vectors on the plane, and s and t are constants. This form is useful for finding points on the plane or for determining if a given vector is parallel to the plane.
Two equations are needed to represent a plane in three-dimensional space. One equation can be used to determine the location of the plane in one direction, and the other equation can determine the location of the plane in another direction. These two equations can be in either cartesian or vector form.