I don't know how to answer this: A vector can be used to represent the path of a drill tip used to bore a deep mine shaft in Sudbury one quarter of the way to the centre of the Earth. Represent the vector using a directed line segment and Cartesian co-ordinates and describe which representation may be more suitable in this situation. Thanks a bunch!

drwls posted this: If x,y,z are coordinates of the mine opening at a point on the surface of the Earth (radius R), then they satisfy x^2 + y^2 + z^2 = R^2.

In this case, the coordinates of the other end of the mine line segment are 0.75 x, 0.75 y, 0.75 z. That may satisy your "Cartesian representation" question.

If the vector from the center of the Earth to the mine opening is R (which should be in boldface type or with an arrow above to denote a vector), then (3/4) R is a vector denoting the tunnel line segment.

The second format appears more suitable.

My issue: Why is it 0.75? How did you get this, and plus if it is one quarter wouldn't be it 0.25? I am just looking for a little clarification, thanks!

Yes, the vector representing the mine shaft is 1/4 of the radius vector. My error.

The value of 0.75 represents three-quarters (or 75%) of the total distance from the center to the mine opening, not one quarter. This is because the mine shaft is described as being bored "one quarter of the way to the center of the Earth," implying that it does not actually reach the center.

To calculate the coordinates of the other end of the mine line segment as (0.75x, 0.75y, 0.75z), the value of 0.75 is used as a scaling factor. By scaling the original Cartesian coordinates (x, y, z) by 0.75, you're effectively moving three-quarters of the distance towards the center from the surface point.

If the vector from the center of the Earth to the mine opening is R, then (3/4) R is used to represent the tunnel line segment. The (3/4) factor indicates that the vector has been scaled by three-quarters, which corresponds to moving one-quarter of the total distance from the mine opening to the center of the Earth.

So, in summary, the value of 0.75 is used to represent the three-quarters distance from the surface point towards the center, and it arises from the given information that the mine shaft is bored one quarter of the way to the center of the Earth.

The value 0.75 is used because the problem states that the drill tip is one quarter of the way to the center of the Earth. If we take the distance from the surface of the Earth to the center as the full distance, then one quarter of that distance would be 0.25 (or 1/4). However, the problem is asking for the coordinates of the other end of the mine line segment, not just the portion of the distance traveled.

Using the Pythagorean theorem, we know that the sum of the squares of the coordinates (x, y, z) will equal the radius of the Earth squared (R^2).

In this case, if we let (x, y, z) represent the coordinates of the mine opening at the surface of the Earth, then x^2 + y^2 + z^2 = R^2.

To represent one quarter of the distance traveled, we can multiply each coordinate (x, y, z) by 0.75. So the coordinates of the other end of the mine line segment become (0.75x, 0.75y, 0.75z).

This representation using Cartesian coordinates allows us to easily understand the relationship between the starting point (mine opening) and the end point (other end of the mine line segment) in a three-dimensional space. It simplifies the calculation and visualization of the path of the drill tip.

Therefore, the representation of the vector using Cartesian coordinates, specifically (0.75x, 0.75y, 0.75z), is more suitable and accurate in this situation.