3.6.6 · D53D Geometry
Question bank — Equation of a line in 3D — vector, symmetric, parametric forms
Quick vocabulary refresher (so nothing below is a surprise)
True or false — justify
A line in 3D needs three ingredients: a point, a direction, and a length.
False — a line has infinite extent, so length is never an ingredient. Only a point (anchor) and a direction vector pin it down; the parameter then reaches every point, sliding you copies of along.
The parameter measures the distance walked from the anchor.
False — scales , so it counts copies of . The distance is ; it equals only when is a unit vector ().
The vectors and give two different lines through the same point.
False — , so they point the same way (see Vectors — scalar multiplication & parallel vectors). Direction ratios are unique only up to a nonzero scalar, so both describe the same line.
If a point satisfies the -equation of a parametric line, it lies on the line.
False — the same must satisfy all three of simultaneously. Fitting one coordinate only fixes one ; the other two must then agree with it.
Two lines with the same direction vector must be the same line.
False — they are parallel, but they can be shifted apart (different anchors not on each other). They coincide only if one line's anchor also satisfies the other's equation.
Choosing always lands you on the anchor point .
True — at , , which is exactly the anchor's position vector.
In symmetric form the numerators tell you the direction of the line.
False — numerators like encode the point ; the denominators are the direction ratios (see Direction ratios and direction cosines).
Reversing the direction vector () gives a different line.
False — walking backwards along your arm still traces the same set of points. just re-labels which hits which point; the line is identical.
The equation describes a line only if .
True — if then for every , so you get a single point, not a line. A direction of zero has no "where it heads."
Two symmetric forms with different-looking denominators must be different lines.
False — denominators can be any nonzero scalar multiple. and are the same line.
Spot the error
"Line through with direction is ." Find the mistake.
The -numerator is wrong: , not . Subtracting a negative coordinate flips the sign — it should read .
"Direction has a zero -ratio, so write ." What's illegal?
Division by is meaningless. A zero direction ratio (DR) means is frozen: write and separately.
"From , the point on the line is ." Spot the slip.
The signs are flipped. gives ; gives ; gives . The point is , not .
"To check if lies on , solve : . It works!" Why is this incomplete?
You only tested . Plug into : . It fails, so the point is not on the line — one satisfied coordinate proves nothing.
"Line through and : direction is ." What went wrong?
That's 's position, not the direction. Direction — the displacement between the points, not one of them.
"Symmetric form works for any line." When does it fail?
It fails whenever any of (the direction ratios, DRs) is , because you cannot divide by that DR. You then split off the frozen coordinate as its own equation.
"The two-point form denominators are the only correct direction ratios." Correct this.
They are a valid set, but so is any nonzero multiple. and half of it describe the identical line.
Why questions
Why does "being on the line" force ?
Because the displacement from a fixed line-point to any line-point runs along the line, hence is parallel to — and parallel vectors are scalar multiples of each other.
Why can we swap freely between vector, parametric, and symmetric forms? — show the elimination.
Start from . Parametric: read it coordinate-by-coordinate, . Symmetric: solve each for the same — — and equate them, so vanishes. All three forms are one equation in different costumes.
Why must the same appear in all three parametric equations?
One walk along the line is one value of ; it moves together. Different 's for different coordinates would describe a point off the line.
Why does a zero direction ratio freeze a coordinate rather than break the line?
If then for every — the -coordinate simply never changes as you walk. The line lives entirely in the plane .
Why is direction (not the anchor) what decides if two lines are parallel?
Parallelism is about heading the same way, which is captured purely by the direction vectors — see Angle between two lines in 3D. Anchors only decide where the lines sit, not their tilt.
Why do we even bother eliminating to get symmetric form?
is scaffolding — it doesn't describe the line's shape. Removing it exposes the point and direction directly in one clean chain of equalities, ideal for reading off DRs.
Why can a scalar multiple of replace without changing the line?
Scaling only rescales how far one unit of carries you (the distance stretches), but the set of reachable points — forwards and backwards, all distances — is unchanged.
Edge cases
What line do you get if the direction vector is ?
Not a line — every gives , so it collapses to the single point . A direction is mandatory.
What does symmetric form look like when two direction ratios are zero, say ?
The line is parallel to the -axis: write and , with free. Only the ratio survives.
Can a "line through two points" ever fail to give a line?
Yes — if , then , so there is no direction and no line, just the shared point. The two points must be distinct.
If all three DRs (direction ratios) are zero in a two-point setup, what happened geometrically?
The two "points" are actually the same point (), so no direction exists. Every difference etc. vanished — a degenerate, non-line case.
Along the line, as and , what happens?
You march off to infinity in opposite directions along the same straight track. Both tails belong to the line — that's why ranges over all of , not just .
When does a point sit between the two defining points and ?
Exactly when its parameter satisfies in ; is , is , and values outside lie beyond the segment.
Recall One-line survival check for every trap (click to reveal)
Before trusting any line answer, ask: (1) Did I rewrite numerators as to get signs right? (2) Did I check the same in all three coordinates? (3) Did any DR hit zero (freeze that coordinate)? (4) Is my "direction" actually , not ? (5) Am I confusing with distance (remember distance )? Five questions kill the common mistakes.