Vectors

How do you find the vector equation of a line in three dimensions?

Quick answer

To find the vector equation of a line in three dimensions, identify a known point on the line with position vector a and a direction vector d. Write the equation as r = a + t * d, where r is any point on the line and t is a scalar parameter.

The topic explained

The basic form of a line in space

In 2D, you describe a line with a slope and an intercept. In 3D space, lines do not have a single slope value. Instead, a line needs an anchor point and a direction vector to show where it points.

The standard vector equation of a line in three dimensions is r = a + t * d. Here, a = (x0, y0, z0) is the position vector of a known point on the line, d = (d1, d2, d3) is the direction vector along the line, and t is a real scalar parameter. As t changes, the point r moves along the entire line.

Finding the line through two points

If you are given two points, A(1, 2, 3) and B(4, 0, 5), pick one point as your anchor. Let a = (1, 2, 3).

Next, find the direction vector d by subtracting A from B: d = B - A = (4 - 1, 0 - 2, 5 - 3) = (3, -2, 2). The line equation is r = (1, 2, 3) + t(3, -2, 2). In parametric form, this is x = 1 + 3t, y = 2 - 2t, and z = 3 + 2t.

Testing points and line relationships

To check if a point lies on a line, substitute its x, y, and z values into the parametric equations. Solve for t in each component. If all three components give the exact same value for t, the point lies on the line.

Two lines in 3D can be parallel, intersecting, coincident, or skew. Parallel lines have direction vectors that are scalar multiples of each other. Skew lines are not parallel and never intersect because they lie in different planes. To check for intersection, equate the components of both lines using two different parameters, solve two equations, and test the values in the third.

Free printable worksheet

3D Line Vector Equations Practice

For Grade 12 math students. Solve each problem by finding vector equations, testing points, or classifying pairs of lines in space.

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3D Line Vector Equations PracticeName:

Write the vector equation of the line passing through point P(2, -1, 4) in the direction of vector d = (3, 5, -2).
Find a vector equation for the line passing through points A(1, 0, 3) and B(4, 2, -1).
Determine whether the point Q(7, 3, 5) lies on the line r = (1, 1, 1) + t(3, 1, 2).
Determine whether the point S(4, -2, 1) lies on the line r = (2, 0, 3) + t(1, -1, -2).
State whether the lines r1 = (1, 2, 3) + s(2, -4, 6) and r2 = (0, 5, -1) + t(-1, 2, -3) are parallel, intersecting, or skew.
Find the point of intersection between line 1: r1 = (1, 0, 1) + s(1, 1, 0) and line 2: r2 = (0, 2, 3) + t(2, -1, -2).

Common questions

Why cannot we use the slope-intercept form y = mx + b in three dimensions?

In three dimensions, an equation with just two variables like y = mx + b defines an entire plane, not a single line. A line needs two independent constraints or a vector equation with a single parameter to constrain motion to one dimension.

What is the difference between parallel and skew lines in space?

Parallel lines have direction vectors that are scalar multiples of each other and lie in the same plane. Skew lines are not parallel, do not intersect, and do not lie in the same plane.

Can the vector equation of a line look different and still be correct?

Yes. You can use any point on the line as the anchor vector a, and any nonzero scalar multiple of the direction vector d. As long as the direction is proportional and the point sits on the line, the equation represents the same line.

How do you convert a vector equation to symmetric form?

Solve each parametric equation (x = x0 + t*d1, y = y0 + t*d2, z = z0 + t*d3) for the parameter t. Setting the expressions equal gives (x - x0)/d1 = (y - y0)/d2 = (z - z0)/d3, provided none of the direction components equal zero.

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