Final answer:
The question involves creating a triangle in a three-dimensional space where vertex C lies on the yz-plane within a circle of radius 1 centered at the origin and z is positive. The analytical representation of vectors in the xyz-coordinate system helps to understand the position of C in relation to the circle equation y^2 + z^2 = 1.
Step-by-step explanation:
The question is asking to consider a triangle in a three-dimensional space with given vertices A(1,0,0), B(0,1,0), and C(x, y, z). Vertex C lies on a circle of radius 1 on the yz-plane, and z is positive.
Since C is on the yz-plane, its x-coordinate is 0. The circle equation on the yz-plane, centered at the origin and with radius 1, is y2 + z2 = 1. As z is positive, the solution for z is the positive square root of 1 - y2.
When we think of a triangle in classical geometry, we consider it as a figure with three sides and three angles, the sum of which is 180 degrees, lying on a plane.
However, in this three-dimensional context, the vertices not only span across the plane but also have depth due to the z-coordinate.
In terms of vectors and their components, as summarized in FIGURE 3.24, vector A with its tail at the origin in an xyz-coordinate system will have components along the x, y, and z axes.
Ax, Ay, and Az in this context represent these components, and they can form a right triangle in their respective planes.
Similar to how vectors and their components work, the points A, B, and C and their coordinates can represent a vector from the origin.