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ezsmith
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Homework Statement
For each part, find the cartesian equation of the plane through the given points.
(1,0,3), (2,-4,3),(4,-1,2)
The Attempt at a Solution
No attempt. Dunno how to do :(
HallsofIvy said:Haven't been paying attention in class?:tongue: Use two pairs of those points to get two vectors in the plane: the vector from [itex](x_0, y_0, z_0)[/itex] to [itex](x_1, y_1, z_1)[/itex] is [itex](x_1- x_0)\vec{i}+ (y_1- y_0)\vec{j}+ (z_1- z_0)\vec{k}[/itex].
The cross product of two vectors is perpendicular to the two vectors so perpendicular to the plane they lie in.
The equation of a plane with normal vector [itex]A\vec{i}+ B\vec{j}+ C\vec{k}[/itex] containing point [itex](x_0, y_0, z_0)[/itex] is [itex]A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0[/itex].
The Cartesian equation of the plane through the given points is a mathematical representation of the plane using the coordinates of the points. It can be written in the form ax + by + cz = d, where a, b, and c are the coefficients of the variables x, y, and z, and d is a constant.
To find the Cartesian equation of a plane through three given points, you can use the formula (x - x1)(y2 - y1) - (x2 - x1)(y - y1) = (x - x1)(z2 - z1) - (x2 - x1)(z - z1), where (x1, y1, z1), (x2, y2, z2), and (x, y, z) are the coordinates of the three points. This formula can be rearranged to get the Cartesian equation of the plane.
Yes, the Cartesian equation of a plane can be written in different forms. One common form is the point-normal form, which is written as (x - x0) · n = 0, where (x0, y0, z0) is a point on the plane and n is the normal vector of the plane. Another form is the intercept form, which is written as x/a + y/b + z/c = 1, where a, b, and c are the intercepts of the plane on the x, y, and z axes, respectively.
Yes, the Cartesian equation of a plane can be used to find the distance between a point and the plane. The formula for this is |ax0 + by0 + cz0 + d| / √(a2 + b2 + c2), where (x0, y0, z0) is the coordinates of the point and a, b, and c are the coefficients of the variables in the Cartesian equation of the plane.
Yes, the Cartesian equation of a plane can be used in three-dimensional geometry. It is a fundamental equation that is used to describe and analyze planes in three-dimensional space. It is also used in many other fields, such as physics, engineering, and computer graphics.