What is the projection of the point on the xy-plane? (x,y,z)=(,) What is the projection of the point on the yz-plane? (x,y,z)=(,) What is the projection of the point on the xz-plane? (x,y,z)=(,) Draw a rectangular box with the origin and (3,5,6) as opposite vertices and with its faces parallel to 3,5,6 What is the projection of the point on the xy-plane? (x,y,z) What is the projection of the point on the yz-plane? x,y,z What is the projection of the point on the xz-plane? x,y,z= Draw a rectangular box with the origin and (3,5,6) as opposite vertices and with its faces parallel to
The Correct Answer and Explanation is:
Correct Answers:
- Projection of (3, 5, 6) on the xy-plane: (x, y, z) = (3, 5, 0)
- Projection of (3, 5, 6) on the yz-plane: (x, y, z) = (0, 5, 6)
- Projection of (3, 5, 6) on the xz-plane: (x, y, z) = (3, 0, 6)
Explanation of Projections and the Rectangular Box
In three-dimensional coordinate geometry, the projection of a point onto a plane can be visualized as the “shadow” that the point casts on that plane when a light source is positioned infinitely far away, perpendicular to the plane. Formally, it is the point where a line, drawn from the original point perpendicular to the plane, intersects the plane.
The coordinate planes are defined by one coordinate being held constant at zero.
- The xy-plane is the set of all points where the z-coordinate is zero.
- The yz-plane is the set of all points where the x-coordinate is zero.
- The xz-plane is the set of all points where the y-coordinate is zero.
To find the projection of a general point P(x, y, z) onto one of these planes, the coordinate not in the plane’s name is set to zero.
Applying this principle to the specific point (3, 5, 6):
- Projection onto the xy-plane: The line perpendicular to the xy-plane is parallel to the z-axis. To find the projection, the z-coordinate is set to zero, while the x and y-coordinates are preserved. Thus, the projection of (3, 5, 6) onto the xy-plane is (3, 5, 0).
- Projection onto the yz-plane: For the yz-plane, the x-coordinate must be zero. The y and z-coordinates remain unchanged. Therefore, the projection of (3, 5, 6) onto the yz-plane is (0, 5, 6).
- Projection onto the xz-plane: For the xz-plane, the y-coordinate is set to zero. The x and z-coordinates are preserved. This gives the projection of (3, 5, 6) onto the xz-plane as (3, 0, 6).
A rectangular box with the origin (0, 0, 0) and the point (3, 5, 6) as opposite vertices has its edges aligned with the coordinate axes. The eight vertices of this box are all the possible combinations of the coordinate values (0, 3), (0, 5), and (0, 6). The vertices are (0,0,0), (3,0,0), (0,5,0), (0,0,6), (3,5,0), (3,0,6), (0,5,6), and (3,5,6). Notice that the three projections calculated above—(3,5,0), (0,5,6), and (3,0,6)—are themselves vertices of this box, lying on the faces that are part of the coordinate planes.
