The illustration at right shows a rectangular box, three of whose edges coincide with the coordinate axes, and one of whose vertices is (3, 6, 4). Give an equation for the plane containing the front face of the box. Do the same for the top and right faces.
the illustration is a rectangular box sitting on the x and y plane and one of the corners is on (0,0,0). the far corner of the box is (3,6,4).
not sure which face is the front, but clearly the point (3,6,4) lies on the plane x=3, and similarly for y and z.
To find the equation for each face of the rectangular box, we need to determine the points that lie on each face. Given that three edges of the box coincide with the coordinate axes and one of the vertices is (3, 6, 4), we can determine the other points on each face using the given information.
1. Front Face:
The front face of the box is parallel to the xy-plane, so all the y-values remain constant on this face. Since one of the vertices is (3, 6, 4), we can find two additional points on the front face by varying the x-coordinate while keeping the y and z-coordinates constant. For example, if we choose (4, 6, 4) and (3, 6, 5) as the other two points, we can find the equation of the plane containing the front face.
To find the equation of the plane containing the front face, we can use the formula for a plane:
Ax + By + Cz + D = 0
Substituting the coordinates of the three points, we get:
4A + 6B + 4C + D = 0 --- (1)
3A + 6B + 5C + D = 0 --- (2)
3A + 6B + 4C + D = 0 --- (3)
We have a system of linear equations with four variables (A, B, C, D). By solving this system, we can obtain the values of A, B, C, and D, and hence determine the equation of the plane containing the front face.
2. Top Face:
The top face of the box is parallel to the xz-plane, so all the y-values remain constant on this face. By varying the x-coordinate while keeping the y and z-coordinates constant, we can find two additional points on the top face. For example, if we choose (3, 7, 4) and (4, 6, 4) as the other two points, we can find the equation of the plane containing the top face.
Using the formula for a plane (Ax + By + Cz + D = 0) and substituting the coordinates of the three points, we get:
3A + 7B + 4C + D = 0 --- (4)
4A + 6B + 4C + D = 0 --- (5)
3A + 6B + 4C + D = 0 --- (6)
By solving this system of linear equations, we can obtain the values of A, B, C, and D, and hence determine the equation of the plane containing the top face.
3. Right Face:
The right face of the box is parallel to the yz-plane, so all the x-values remain constant on this face. By varying the y-coordinate while keeping the x and z-coordinates constant, we can find two additional points on the right face. For example, if we choose (3, 6, 5) and (3, 7, 4) as the other two points, we can find the equation of the plane containing the right face.
Using the formula for a plane (Ax + By + Cz + D = 0) and substituting the coordinates of the three points, we get:
3A + 6B + 5C + D = 0 --- (7)
3A + 7B + 4C + D = 0 --- (8)
3A + 7B + 5C + D = 0 --- (9)
By solving this system of linear equations, we can obtain the values of A, B, C, and D, and hence determine the equation of the plane containing the right face.
By following these steps and solving the systems of linear equations, we can find the equations of the planes containing the front, top, and right faces of the rectangular box.