Problem 2 - Essential Calculus Early Transcendentals

Question

Sketch the points \((0,5,2),(4,0,-1),(2,4,6),\) and \((1,-1,2)\) on a single set of coordinate axes.

Step-by-Step Solution

Answer

Plot each point by locating its position along x, y, and z axes; visualize them on a 3D plane.

Step 1: Understand the 3D Coordinate System

In a 3D coordinate system, each point is represented by three coordinates \(x, y, z\). These coordinates indicate the position along the X-axis, Y-axis, and Z-axis respectively. Before plotting the points, make sure you understand how to navigate these axes.

Step 2: Identify Each Point's Coordinates

List the coordinates of each point:- \(0,5,2\) where \(x = 0, y = 5, z = 2\).- \(4,0,-1\) where \(x = 4, y = 0, z = -1\).- \(2,4,6\) where \(x = 2, y = 4, z = 6\).- \(1,-1,2\) where \(x = 1, y = -1, z = 2\).

Step 3: Sketch the Axes

Draw the three axes: X, Y, and Z. The X-axis is typically horizontal, the Y-axis is vertical, and the Z-axis comes out of the page or screen.

Step 4: Plot the Points

On your 3D graph:- Start by plotting each point one at a time using its coordinates.- Locate the position of each coordinate along the respective axis.- Mark the points: - Point A at \(0, 5, 2\): On the y-axis go up to 5, z-axis go up to 2; no need to move along the x-axis. - Point B at \(4, 0, -1\): Move 4 units along x-axis, no movement on y-axis, then down to -1 on the z-axis. - Point C at \(2, 4, 6\): Go 2 units on x-axis, 4 units up on y-axis, and finally move to 6 on the z-axis. - Point D at \(1, -1, 2\): One unit on x-axis, move down to -1 on y-axis, and move up to 2 on the z-axis.

Step 5: Verify the Points

Check each point to ensure that it is plotted correctly by reviewing its coordinates and the corresponding location on the graph.

Key Concepts

Plotting Points

Plotting points in a 3D coordinate system can initially feel a bit tricky, but with practice, it becomes straightforward. In 3D space, each point is identified by three numbers: the x-coordinate, the y-coordinate, and the z-coordinate. These coordinates show where the point lies along the respective X, Y, and Z axes.

To plot a point:

• Start with the x-coordinate. Move horizontally along the X-axis to locate this first position.
• Next, use the y-coordinate to adjust vertically along the Y-axis.
• Finally, follow the z-coordinate to move directly outwards or inwards, perpendicular to the XY-plane, aligning with the Z-axis.

When you combine all these movements, you position the point in 3D space. Keep the axes orientations in mind: the X-axis runs horizontally, the Y-axis vertically, and the Z-axis pops out towards you or into the page.

3D Graph Sketching

Creating a 3D graph involves meticulously drawing out the three coordinate axes and positioning each point using their coordinates. Start with the X, Y, and Z axes. Typically, artists depict the X-axis horizontally and the Y-axis vertically when sketching on paper. The Z-axis diverges from the XY-plane, coming either out of or into the paper.

Once the axes are drawn:

• Select one point at a time to plot.
• Identify how much you must travel along each axis based on the coordinates.
• Carefully plot each point, using the logical order of moving along the X-axis first, then Y, and last the Z-axis.

By following this method, you can accurately achieve a smooth and precise 3D graph. While sketching visually, maintain consistency in the proportions and orientations of all axes for clarity.

Coordinate Axes

The 3D coordinate system is built on three axes: X, Y, and Z. They serve the primary role of enabling precise location placement in space.

**X-axis**: Represents the horizontal plane. Moving along this axis involves left-to-right motion in the XY-plane.

**Y-axis**: Illustrates the vertical direction. Moving up or down along this axis corresponds to typical vertical motion.

**Z-axis**: Allows representation of depth. It allows movements outward from the plane of the paper or screen (positive direction) or inward (negative direction).

On graph paper or digital interfaces, these axes help visualize complex 3D shapes and configurations. Understanding how to navigate between these axes is crucial to manipulate and understand spatial problems effectively.