(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 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:
Identifying Key Projections
I’ve started by clearly defining the core task: pinpointing the projections of the point (3, 5, 6) onto the xy, yz, and xz planes. This sets the foundation for a focused solution. Next, I’ll figure out what each projection implies in terms of coordinate changes.
Exploring Plane Projections
I’m now focusing on mapping the point (3, 5, 6) onto each plane. It’s clear that the xy-plane projection will keep the x and y values, setting z to zero. Similar logic applies to the yz and xz planes. This will result in three distinct coordinate triplets, which I’ll present. After that I plan to craft a 300-word explanation that echoes standard textbook style.
Refining Coordinate Projections
I’ve determined the projections: (3, 5, 0), (0, 5, 6), and (3, 0, 6) onto the xy, yz, and xz planes, respectively. Now, I’m synthesizing these findings into a detailed explanation. My goal is to adopt a textbook style, avoiding first-person pronouns, and ensuring a precise 300-word account of the projection process. I will now create an image of a box with the coordinates in mind, and the explanation will explain the shape based on these points.
Calculating Coordinate Projections
I’ve now finalized the solution and crafted the formal explanation. The projections are: xy-plane (3, 5, 0), yz-plane (0, 5, 6), and xz-plane (3, 0, 6). The 300-word textbook-style explanation is complete, focusing on the geometric interpretation of these projections within the context of a rectangular box defined by the origin and the point (3, 5, 6).
Refining Plane Visualization
I’ve crafted a comprehensive explanation for the projections onto each plane, adhering to the 300-word limit and textbook style. The rectangular box and shadow analogies are integrated for clarity. Further review is planned to ensure precision and conciseness, readying the final output.
Synthesizing Geometric Concepts
I’ve crafted an initial solution and written a detailed explanation. The projections are (3, 5, 0), (0, 5, 6), and (3, 0, 6). The explanation, around 300 words, uses the rectangular box analogy to elucidate the projections, with an emphasis on textbook style and third-person narrative. I’ll now refine the explanation to solidify its clarity.
Formulating the Final Response
I’ve crafted and finalized the complete answer, encompassing both the immediate solutions and the 300-word explanatory section. The format is now precisely as requested: initial answers followed by a detailed, textbook-style explanation. The rectangular box and shadow analogies are integrated throughout to clarify the projection concepts, maintaining a focused and formal tone.
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What is the projection of the point on the xy-plane? (x,y,z)=(3,5,0)
What is the projection of the point on the yz-plane? (x,y,z)=(0,5,6)
What is the projection of the point on the xz-plane? (x,y,z)=(3,0,6)
In a three-dimensional Cartesian coordinate system, the projection of a point onto a plane is the point on the plane that is geometrically closest to the original point. This can be visualized as the “shadow” cast by the point onto the plane from a light source positioned infinitely far away along the axis perpendicular to that plane.
Consider the point P(3, 5, 6). To conceptualize its projections, one can imagine a rectangular box, or cuboid, with one vertex at the origin O(0, 0, 0) and the diagonally opposite vertex at P(3, 5, 6). The edges of this box are aligned with the coordinate axes, extending 3 units along the x-axis, 5 units along the y-axis, and 6 units along the z-axis. The other vertices of this box represent the various projections of the point P.
The projection of P(3, 5, 6) onto the xy-plane is found by setting its z-coordinate to zero. The xy-plane is defined by the equation z = 0. Dropping a perpendicular line from P straight down to this plane results in the point (3, 5, 0). This point retains the original x and y coordinates but lies on the “floor” of the coordinate system. In the context of the rectangular box, this projection is the vertex that lies on the xy-plane directly below P.
The projection of P(3, 5, 6) onto the yz-plane is found by setting its x-coordinate to zero. The yz-plane is defined by the equation x = 0. Projecting P perpendicularly onto this plane yields the point (0, 5, 6). This is analogous to casting a shadow onto the “back wall” of the coordinate system. This projection is another vertex of the described rectangular box.
Finally, the projection of P(3, 5, 6) onto the xz-plane is found by setting its y-coordinate to zero. The xz-plane is defined by the equation y = 0. The resulting projection is the point (3, 0, 6), which can be visualized as the shadow on the “side wall” of the coordinate system and is also a vertex of the box.
