Consider the point (4, 5, 6). What is the projection of the point on the xy-plane? (x, y, z) = (4, 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) = (4, 0, 6). Draw a rectangular box with the origin and (4, 5, 6) as opposite vertices and with its faces parallel to the
Find the length of the diagonal of the box.
The Correct Answer and Explanation is:
Let’s break down the problem step-by-step, and then calculate the length of the diagonal of the rectangular box formed by the origin (0,0,0)(0,0,0) and the point (4,5,6)(4,5,6) as opposite vertices.
Step 1: Understand the projection points
- Projection on the xy-plane: To project a point onto the xy-plane, we set the zz-coordinate to 0 but keep xx and yy the same.
So, (4,5,6)→(4,5,0)(4, 5, 6) \to (4, 5, 0). - Projection on the yz-plane: To project on the yz-plane, set x=0x=0, keep yy and zz.
So, (4,5,6)→(0,5,6)(4, 5, 6) \to (0, 5, 6). - Projection on the xz-plane: To project on the xz-plane, set y=0y=0, keep xx and zz.
So, (4,5,6)→(4,0,6)(4, 5, 6) \to (4, 0, 6).
Step 2: Visualizing the rectangular box
The box has:
- One vertex at the origin: (0,0,0)(0,0,0)
- Opposite vertex at (4,5,6)(4, 5, 6)
The edges of the box align with the coordinate axes, so its sides are lengths:
- Along xx: 4 units
- Along yy: 5 units
- Along zz: 6 units
Step 3: Find the length of the diagonal
The diagonal is the line segment connecting the two opposite vertices (0,0,0)(0,0,0) and (4,5,6)(4,5,6).
We use the 3D distance formula: d=(x2−x1)2+(y2−y1)2+(z2−z1)2d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}
Substitute: d=42+52+62=16+25+36=77d = \sqrt{4^2 + 5^2 + 6^2} = \sqrt{16 + 25 + 36} = \sqrt{77}
Final answer:
77≈8.775\boxed{ \sqrt{77} \approx 8.775 }
Explanation (300+ words):
In three-dimensional geometry, the concept of projection and distance between points is fundamental to understanding shapes and spatial relationships. Here, we are given a point (4,5,6)(4, 5, 6) in 3D space, and asked to determine its projections on the coordinate planes and find the length of the diagonal of a rectangular box defined by this point and the origin.
Projections on coordinate planes are straightforward. A projection onto a plane means dropping the point perpendicularly onto that plane. For the xy-plane, this means eliminating the zz-coordinate; for the yz-plane, eliminate the xx-coordinate; and for the xz-plane, eliminate the yy-coordinate. This yields the points (4,5,0)(4, 5, 0), (0,5,6)(0, 5, 6), and (4,0,6)(4, 0, 6) respectively, as expected.
Next, the rectangular box is formed by the origin and the point (4,5,6)(4, 5, 6) as opposite vertices. Since the faces of the box are parallel to the coordinate planes, its edges lie along the xx, yy, and zz axes. The side lengths of the box correspond to the differences in the coordinates between these two points, which are 4, 5, and 6 units along the xx, yy, and zz directions respectively.
To find the diagonal, we apply the three-dimensional version of the Pythagorean theorem. The diagonal length is the distance from the origin to the opposite corner, which is found by taking the square root of the sum of the squares of the side lengths: d=42+52+62=16+25+36=77.d = \sqrt{4^2 + 5^2 + 6^2} = \sqrt{16 + 25 + 36} = \sqrt{77}.
The numerical value is approximately 8.775 units.
This calculation generalizes well to any rectangular box in 3D space. The diagonal is the longest straight line inside the box and can be found easily with the distance formula. Understanding these spatial relationships is useful in many fields such as computer graphics, physics, engineering, and architecture where 3D modeling and distances need to be accurately computed.
If you’d like, I can also help you with the diagram or a 3D visualization of this box
