The first octant in 3-space has points with three positive coordinates. We must use precise terminology to describe the location of points and vectors in 3-space. For the following question, you need to remember the following key points There are 4 quadrants There are 8 octants Quadrant Quadrant y=0 2 1 x=0 Origin Quadrant Quadrant (0,0,0) 3 4 The 3D Cartesian plane has the following axis z positive positive positive z Octant 1 Y-coordinate X-coordinate 4. Describe the axis, plane, or octant on/in which each is point is located. Use the proper terminology. (10 marks) (1,2,3) octant (4,0,-11) plane (-8,0,0) x-axis (5,2, -7) Front right octant (0, -5,9) plane (0,0,6) z-axis (-4, -3, -7) Back Left octant (-3,2,1) right top octant (0,7,0) Positive -axis (0,0,0)
The Correct Answer and Explanation is:
To correctly classify each point in three-dimensional space, we must use precise terminology related to axes, planes, and octants. The 3D Cartesian coordinate system consists of eight octants, determined by the signs of the x, y, and z coordinates. Points that lie directly on an axis or a plane are classified accordingly.
- (1,2,3) – This point is located in the first octant because all three coordinates are positive. The first octant contains points where x > 0, y > 0, and z > 0.
- (4,0,-11) – This point lies on the xz-plane. Since the y-coordinate is zero, the point belongs to this plane, which separates the front and back portions of 3-space.
- (-8,0,0) – This point lies on the x-axis. Since both the y and z coordinates are zero, it is directly on the x-axis, extending leftward due to the negative x-value.
- (5,2,-7) – This point is in the fourth octant, commonly referred to as the front right octant. Here, x and y are positive while z is negative.
- (0,-5,9) – This point lies on the yz-plane. Since the x-coordinate is zero, the point resides in the vertical plane spanning y and z values.
- (0,0,6) – This point is on the z-axis. Because both x and y are zero, the coordinate is positioned directly along the z-direction.
- (-4,-3,-7) – This point is in the seventh octant, referred to as the back left octant. Here, all three coordinates are negative.
- (-3,2,1) – This point is in the second octant, which is often called the right top octant. The x-value is negative while the y and z values are positive.
- (0,7,0) – This point is located on the y-axis. Because the x and z values are zero, it is positioned along the vertical axis.
- (0,0,0) – This point is at the origin, the central reference point where all coordinates are zero.
In three-dimensional space, categorization helps in visualization and problem-solving. Axes serve as fundamental reference lines, and planes divide the space into distinct regions. Understanding these concepts is essential in geometry and physics.
