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Math101learn.math101.caLines in Three Dimensions
A line in three-dimensional space is determined by one point and one nonzero direction vector.
A spatial line is every point reached by starting at one known point and moving any scalar amount along a fixed direction.
Vector equation
If a line passes through point $P_0(x_0,y_0,z_0)$ with nonzero direction vector
then its vector equation is
The parameter $t$ moves forward or backward along the line. Different values produce different points.
Parametric equations
The vector equation is equivalent to
These three equations share the same parameter. Treating them with separate unrelated parameters would describe a larger set than one line.
Symmetric form
When $a,b,c$ are all nonzero, eliminate $t$:
If a direction component is zero, use a fixed coordinate instead. For example, $a=0$ means $x=x_0$ along the entire line, so division by $a$ is not allowed.
Worked example: write all forms
At $t=0$, the line contains the required point.
Line through two points
For points $A$ and $B$, a direction is
Then use either point as $\vec r_0$. Reversing the direction or multiplying it by any nonzero scalar gives the same geometric line with a different parameterization.
Checking whether a point lies on a line
Substitute the point's coordinates into the parametric equations. A single value of $t$ must satisfy all three equations.
If different coordinates require different parameter values, the point does not lie on the line.
Relationships between two lines
In three dimensions, two lines may:
- intersect at one point;
- be parallel and distinct;
- coincide;
- be skew, meaning nonparallel and nonintersecting.
Skew lines are possible because they can pass at different levels without sharing a plane.
Testing an intersection
Give the lines different parameters, such as $s$ and $t$, and equate corresponding coordinates. Solve the resulting system, then verify all three coordinate equations.
Consistent parameters give an intersection point. An inconsistent system with nonparallel directions indicates skew lines.
Angles between lines
The acute angle between lines can be found from direction vectors:
The absolute value selects the acute angle between unoriented lines. Without it, the vector directions may give the supplementary angle.
Distance and geometry
Distances involving skew lines or a point and a line use perpendicular components, projections, or cross products. The shortest connector must be perpendicular to the relevant direction vectors.
Draw a spatial diagram and state which distance is required before selecting a formula.
Common mistakes
Using the position point as the direction. A line needs both, and they serve different roles.
Assigning a different parameter to each coordinate. One line uses one shared parameter.
Dividing by a zero direction component in symmetric form. Write that coordinate as constant.
Calling all nonintersecting lines parallel. In space they may be skew.
Stopping after two coordinate equations agree. Verify the third.
Quick self-check
- Is the direction vector nonzero?
- Does $t=0$ recover the stated point?
- Do vector, parametric, and symmetric forms describe the same line?
- Does a tested point use one consistent parameter?
- For two lines, were parallel, coincident, intersecting, and skew cases distinguished?
- Is the angle or distance interpretation based on direction vectors?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Which parametric equations describe the line through (1, −2, 3) parallel to ⟨2, 1, −1⟩?
- Start at (1, −2, 3).
- Add t⟨2, 1, −1⟩ component by component.
End of lesson
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