Let a,b∈R3 be fixed, distinct vectors, and let x:R→R3 be the function defined by x(t)=a+et(b−a).

Which of the following is a geometric description of the curve parametrised by x? There is one correct answer.

[Note: a line segment is a portion of a line with both a start and end point; a ray is a portion of a line with a start but no end point.]

Question

Let a,b∈R3 be fixed, distinct vectors, and let x:R→R3 be the function defined by x(t)=a+et(b−a).

Which of the following is a geometric description of the curve parametrised by x? There is one correct answer.

[Note: a line segment is a portion of a line with both a start and end point; a ray is a portion of a line with a start but no end point.]

BlackTom AI · Accepted Answer

First, restate the problem in your own words to ensure understanding: we have x(t) = a + e^t(b - a) with fixed distinct vectors a and b in R^3, and t ranges over all real numbers. We want a geometric description of the curve traced by x as t varies.

Consider what e^t can do. Since e^t > 0 for all real t, the scalar factor multiplying (b - a) is always positive. This means every point x(t) lies on the line through a and b, because x(t) = a plus a positive multiple of the direction from a to b. But because e^t > 0, we never get back to a itself (the scalar is never 0), so the curve does not include the point a. When t increases, e^t grows without bound, pushing x(t) further along the same line past b and beyond; when t = 0, e^t = 1 and x(0) = a + (b - a) = b, so the curve passes through b.

Now evaluate the given option: "The ray that begins at a (but excludes a) and passes through b." A ray is defined as starting at a given point and extending infinitely in one direction along a line. Here, the curve starts at a (not included) and moves along the line from a toward and beyond b, matching exactly the definition of a ray starting at a and passing through b. Therefore, the description aligns perfectly with the geometry of x(t).

Why other possibilities would be incorrect if they were present (conceptual checks): if one claimed a line segment, that would require the curve to terminate at b, which is not the case since e^t grows without bound as t → ∞. If one claimed a full line through a and b, that would require the curve to include points on the other side of a as t varies; however, because e^t is always positive, we never reach points on the opposite ray from a. And if someone described only the portion from a to b without extending beyond b, that would again constrain the curve to a finite segment, which is incorrect given the exponential growth for large t.

With these considerations, the single accurate geometric description is the ray starting at a (excluding a) and passing through b.

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