91Ó°ÊÓ

When finding the distance from a point to the xy-plane in 3D geometry, you focus on the z-coordinate of the point. Imagine that the xy-plane lies flat like a floor, and any point off this plane will have a height corresponding to its z-value.

For a point \(3, 7, -5\), its height above or below this 'floor' is simply the absolute value of the z-coordinate. Here, the z-coordinate is -5, so the distance is \(|-5| = 5\). This distance can be thought of as how far up or down the point is from the xy-plane.

• The formula for the distance to the xy-plane is \(|z|\).
• This distance focuses solely on vertical displacement.

Knowing just the z-value is sufficient for this calculation.

The yz-plane is like a wall standing upright. Points not on this plane have an x-coordinate that tells us how far they are from the yz-plane.

In the example with the point \(3, 7, -5\), the x-coordinate is 3. Thus, the distance to the yz-plane is \(|3| = 3\).

What happens here is that we effectively ignore changes in y and z since moving along these does not affect how far the point is from the yz-plane.

• The formula for distance to the yz-plane is \(|x|\).
• This considers only horizontal displacement along the x-axis.

This is the simplest dimension to visualize: moving directly left or right away from the yz-plane.

The xz-plane distance draws focus to how high or low a point is relative to this plane. Here, you're checking the y-coordinate for position details.
Using the point \(3, 7, -5\), observe that the y-coordinate is 7. Hence, its distance from the xz-plane is \(|7| = 7\).

Since the xz-plane cuts through space like a vertical wall, distance is measured by how far above or below the wall the point sits, determined exclusively by y.

• The formula for the distance to the xz-plane is \(|y|\).
• This metric considers only vertical displacement along the y-axis, ignoring x and z.

One can imagine climbing a ladder, either up or down, to exit the plane.

Distance to an axis uses two coordinates since an axis is where two planes intersect. Each axis forms a line in 3D space, requiring more precision to measure distance.

X-Axis

For distance from \((3,7,-5)\) to the x-axis, use the formula \[\sqrt{(y-0)^2+(z-0)^2}\]. Plugging in 7 and -5 yields \[\sqrt{49+25} = \sqrt{74}\]. This measures movement in both y and z to the x-axis.

Y-Axis

For the y-axis, calculate with \[\sqrt{(x-0)^2+(z-0)^2}\]. Using 3 and -5, we get \[\sqrt{9+25} = \sqrt{34}\]. Here, x and z are the distances of interest.

Z-Axis

Lastly, for the z-axis, distance is \[\sqrt{(x-0)^2+(y-0)^2}\]. With x=3 and y=7, this gives \[\sqrt{9+49} = \sqrt{58}\]. This involves moving along x and y axes.

• Distances to axes use Pythagorean theorem across two coordinates.
• This accounts for diagonal distance rather than single-axis measurement.

Finding these distances helps in visualizing three-dimensional navigation in relation to coordinate lines.