MAT188H1 Chap.4 Parametric Lines
Parametric Lines
A line needs an anchor and a nonzero direction
A vector description of a line has the form p + td, where p is the position vector of a fixed point, d is a nonzero direction vector, and t ranges over all real numbers. The anchor determines where the line is placed. The direction determines how its points vary.
These roles are different even though p and d are both written as vectors of the same dimension.
The nonzero condition prevents the expression from collapsing to a single point. If d were zero, varying t would leave p unchanged. An anchor at the origin is allowed, but an origin anchor is not required.
A line through the origin consists of all scalar multiples of its direction vector; a translated line adds a fixed position vector to those multiples.
Two points supply a displacement
Given distinct points P and Q, subtract their coordinates to obtain a nonzero direction Q - P. Taking P as the anchor gives P + t(Q - P). At t = 0 the expression returns P, and at t = 1 it returns Q.
These substitutions are useful checks because they connect the constructed expression directly to the information in the question.
Reversing the subtraction gives the opposite direction and still describes the same full line when the parameter ranges over all real values. Likewise, a nonzero scalar multiple of a direction vector can be used instead. The parameter values assigned to individual points then change.
Do not confuse a different speed or orientation in the formula with a different geometric set.
One parameter controls every coordinate
Expanding the vector expression component by component produces parametric equations. In space, x = p₁ + td₁, y = p₂ + td₂ and z = p₃ + td₃ use the same t. A point lies on the line only if a single parameter value satisfies all the equations simultaneously.
Finding a separate value from each coordinate and accepting them independently would test three unrelated conditions.
A zero direction component produces a fixed coordinate. This is a constraint, not a reason to divide by zero. If d₂ is zero, every point on the line has second coordinate p₂. A proposed point with a different second coordinate fails immediately.
If it matches, use another nonzero direction component to determine t and check the remaining coordinates.
Membership is a simultaneous check
To test a candidate point, subtract the anchor and ask whether the resulting displacement is a scalar multiple of the direction. This is the vector version of solving the coordinate equations.
Start with a nonzero component of d, calculate a candidate parameter, then substitute into every coordinate. A mismatch in any coordinate excludes the point, even if the diagram appears to place it near the line.
Keep exact fractions during this test. A rounded parameter may make a genuine equality look slightly false, while broad rounding may hide an actual mismatch.
The final statement should name whether the point belongs and show the common parameter or the failing coordinate. This gives a stronger explanation than a list of unrelated coordinate calculations.
Different formulas can trace the same line
An alternative anchor can be chosen anywhere on the original line. If q = p + ad for some real a, then q + sd = p + (a + s)d.
Because s ranges over all real numbers, the shifted parameter a + s does too. The expression therefore generates the same collection of points. This algebra explains why anchors are not unique.
Parallel direction vectors alone do not establish equal lines: the anchors must also be appropriately related. Two lines may share a direction while sitting in different locations.
To show that two formulas define the same line, compare their nonzero directions and test whether an anchor from one belongs to the other. Both pieces are necessary because they control orientation and placement separately.
A sketch should show the anchor, the direction arrow and the whole line extending through them. Plotting a few parameter values makes the formula concrete, but those samples are not the definition.
Preserve the real parameter domain and the nonzero direction condition whenever you rewrite the answer. They are what make the expression a complete line rather than a restricted path or a single point.
What this chapter covers
- 01
Point and direction form
- 02
Coordinate equations and membership
- 03
Equivalent anchors and directions
Worked example · free
One parameter or three incompatible values?
- 2The coordinate equations are x = 1 + 2t, y = -2 + 3t and z = 4 - t. Both candidates have x = 5, so the first equation requires 5 = 1 + 2t and hence t = 2.
- 2At t = 2, the second coordinate is -2 + 6 = 4 and the third is 4 - 2 = 2. Thus r(2) = (5, 4, 2), exactly the coordinates of A.
- 2B has the same first coordinates but a third coordinate of 1. That third equation would require t = 3, incompatible with the t = 2 already required by x and y.
Key terms
- Direction Vector
- A nonzero vector parallel to a line that specifies how its coordinates vary. Multiplying it by any nonzero real scalar changes the parameter assignment while retaining the full line direction.
- Anchor Point
- A fixed point used to position a line in its vector equation. The anchor is obtained at parameter zero in the chosen description, but another point on the line can replace it.
- Parametric Equations
- Coordinate equations that describe an object through common parameter values. For a line, one real parameter controls all coordinates simultaneously, so their conditions cannot be satisfied using independent values.
- Line Membership
- The property of a point being generated by the defining expression of a line. It requires a single real parameter that reproduces every coordinate, or equivalently a displacement parallel to the direction vector.
Parametric Lines FAQ
Can two correct line equations use opposite direction vectors?
Yes. Replacing d with -d reverses which parameter values reach which points. When the parameter still ranges over all real numbers, every original point remains reachable by changing the parameter sign. Check that the anchor has also been kept on the same line before declaring the equations equivalent.
What should I do when a coordinate equation has no parameter?
Treat that coordinate as fixed. It gives an immediate membership condition that every point on the line must satisfy. Use a coordinate with a nonzero direction component to calculate the parameter, then return to the fixed coordinate and all remaining equations to verify the candidate.
Why do matching directions fail to prove two lines are identical?
The direction says how to move but does not establish a location. Distinct parallel lines can have the same direction vector. Substitute an anchor from one line into the equations for the other. If it belongs and the nonzero directions are scalar multiples, the descriptions trace the same complete set.
Exam move
Construct a line from two distinct points, then recover both points by substitution. Change the anchor and rescale the direction without changing the set. Finally, alter one coordinate of a known point and explain precisely why the altered point fails the common parameter test. These tasks separate construction, equivalence and membership.
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