Question

Describe the vertices and edges of the rectangular parallelopiped with vertex (3, 5, 6) placed in the first octant with one vertex at origin and edges of parallelopiped lie along x, y and z-axes.

Answer 1

The six planes of the parallelopiped are as follows:

Plane OABC lies in the xy-plane. The z-coordinate of every point in this plane is zero. z = 0 is the equation of this xy-plane. Plane PDEF is parallel to xy-plane and 6 unit distance above it. The equation of the plane is z = 6. Plane ABPF represents plane x = 3. Plane OCDE lies in the yz-plane and x = 0 is the equation of this plane. Plane AOEF lies in the xz-plane. The y coordinate of everypoint in this plane is zero. Therefore, y = 0 is the equation of plane

Plane BCDP is parallel to the plane AOEF at a distance y = 5.

Edge OA lies on the x-axis. The x-axis has equation y = 0 and z = 0.

Edges OC and OE lie on y-axis and z-axis, respectively. The y-axis has its equation z = 0, x = 0. The z-axis has its equation x = 0, y = 0. The perpendicular distance of the point P (3, 5, 6) from the x-axis is √(5² +6²) = 61. The perpendicular distance of the point P (3, 5, 6) from y-axis and z-axis are √(3² + 6²) = 45 and √(3² + 5² ) =, respectively.

The coordinates of the feet of perpendiculars from the point P (3, 5, 6) to the coordinate axes are A, C, E. The coordinates of feet of perpendiculars from the point P on the coordinate planes xy, yz and zx are (3, 5, 0), (0, 5, 6) and (3, 0, 6). Also, perpendicular distance of the point P from the xy, yz and zx-planes are 6, 5 and 3, respectively, Fig.12.8.