Consider the point. (1, 3, 4) 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) = )
Consider the point. (1, 3, 4) 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) = )
The Correct Answer and Explanation is:
Answer:
Consider the point. (1, 3, 4)
What is the projection of the point on the xy-plane? (x, y, z) = (1, 3, 0)
What is the projection of the point on the yz-plane? (x, y, z) = (0, 3, 4)
What is the projection of the point on the xz-plane? (x, y, z) = (1, 0, 4)
Explanation
In three-dimensional Cartesian space, the location of any point is given by an ordered triple of coordinates (x, y, z). This space is fundamentally structured by three mutually perpendicular coordinate planes: the xy-plane, the yz-plane, and the xz-plane. The projection of a point onto one of these planes is the location where a perpendicular line from the point intersects the plane. This projected point can be thought of as the shadow cast by the original point onto the plane.
Projection onto the xy-Plane
The xy-plane is the set of all points for which the z-coordinate is zero (z = 0). To project a point (x₀, y₀, z₀) onto this plane, one must find the point on the plane that is closest to it. This is achieved by dropping a perpendicular from the point to the plane. A line perpendicular to the xy-plane is parallel to the z-axis. Along such a line, the x and y-coordinates do not change. Therefore, the projection is found by simply setting the z-coordinate to zero. For the given point (1, 3, 4), its projection onto the xy-plane is (1, 3, 0).
Projection onto the yz-Plane
The yz-plane is defined by the equation x = 0. A line perpendicular to this plane is parallel to the x-axis. To find the projection of a point (x₀, y₀, z₀) onto the yz-plane, the x-coordinate must be set to zero, while the y and z-coordinates are preserved. Applying this to the point (1, 3, 4), its x-coordinate of 1 is replaced with 0, yielding the projection (0, 3, 4).
Projection onto the xz-Plane
Finally, the xz-plane consists of all points where the y-coordinate is zero (y = 0). A line perpendicular to the xz-plane is parallel to the y-axis. Consequently, projecting a point onto this plane involves retaining the original x and z-coordinates and setting the y-coordinate to zero. For the point (1, 3, 4), its projection onto the xz-plane is therefore (1, 0, 4).
