Vector Equation — Line and Plane Forms, Examples

Vector Equation — Line and Plane Forms, Examples

TL;DR

A vector equation describes a line or a plane using position vectors and a parameter, most often the line form ( \vec{r} = \vec{a} + \lambda \vec{b} ) — start at point ( \vec{a} ), then slide along direction ( \vec{b} ). This article covers the line form, the two-point form, the plane forms, what the parameter ( \lambda ) does, how to convert to Cartesian form, and six worked examples.

What Is a Vector Equation?

A vector equation is an equation that uses vectors to describe a geometric object — usually a line or a plane in two or three dimensions. Instead of relating ( x ), ( y ), and ( z ) directly, it relates position vectors (arrows from the origin to a point) using a scalar parameter.

The most common vector equation is the equation of a line, ( \vec{r} = \vec{a} + \lambda \vec{b} ). Here ( \vec{r} ) is the position vector of any point on the line, ( \vec{a} ) is the position vector of one known point, ( \vec{b} ) is a direction vector parallel to the line, and ( \lambda ) is the parameter you vary. The whole subject rests on combining vectors through addition and scalar multiplication.

What Is the Vector Equation of a Line?

The vector equation of a line through a point with position vector ( \vec{a} ), parallel to direction vector ( \vec{b} ), is:

[ \vec{r} = \vec{a} + \lambda \vec{b} ]

where ( \vec{r} ) is the position vector of a general point on the line and ( \lambda ) is a real-number parameter. Read it as a journey: travel from the origin out to point ( \vec{a} ), then move some multiple ( \lambda ) of the direction ( \vec{b} ).

The two-point form

If you're given two points rather than a point and a direction, the direction is the vector from one point to the other. For points with position vectors ( \vec{a} ) and ( \vec{b} ), the direction is ( \vec{b} - \vec{a} ), so:

[ \vec{r} = \vec{a} + \lambda (\vec{b} - \vec{a}) ]

At ( \lambda = 0 ) you sit at ( \vec{a} ); at ( \lambda = 1 ) you land exactly on ( \vec{b} ). Any other value of ( \lambda ) places you somewhere along — or beyond — the segment joining them.

What does the parameter ( \lambda ) actually do?

( \lambda ) is the dial that picks out which point on the line you mean. Each value of ( \lambda ) names exactly one point:

As ( \lambda ) runs over every real number, ( \vec{r} ) sweeps out the entire infinite line. This is exactly the parametric idea: one variable, the whole curve.

What Is the Vector Equation of a Plane?

A plane needs more pinning down than a line, and there are three standard vector equations for it depending on what you're given.

How Do You Convert a Vector Equation to Cartesian Form?

The Cartesian form drops out when you compare components and eliminate ( \lambda ). Take a line ( \vec{r} = \vec{a} + \lambda \vec{b} ) with ( \vec{a} = (x_1, y_1, z_1) ) and ( \vec{b} = (b_1, b_2, b_3) ). Writing ( \vec{r} = (x, y, z) ) component by component gives:

[ x = x_1 + \lambda b_1, \quad y = y_1 + \lambda b_2, \quad z = z_1 + \lambda b_3 ]

Solve each for ( \lambda ) and set them equal:

[ \frac{x - x_1}{b_1} = \frac{y - y_1}{b_2} = \frac{z - z_1}{b_3} ]

That symmetric form is the Cartesian equation of the line. The vector form and the Cartesian form describe the same line — one keeps the parameter, the other eliminates it.

Examples of Vector Equation

Example 1

Find the vector equation of the line through the point ( (1, 2, 3) ) parallel to ( \vec{b} = (2, -1, 4) ).

The base point gives ( \vec{a} = (1, 2, 3) ) and the direction is ( \vec{b} = (2, -1, 4) ). Drop them into ( \vec{r} = \vec{a} + \lambda \vec{b} ):

[ \vec{r} = (1, 2, 3) + \lambda(2, -1, 4) ]

Final answer: ( \vec{r} = (1, 2, 3) + \lambda(2, -1, 4) ).

Example 2

Find the vector equation of the line through A( (1, 0, 2) ) and B( (4, 3, 2) ).

Correct. The direction between two points is ( \vec{b} - \vec{a} ):

[ \vec{r} = (1, 0, 2) + \lambda((4, 3, 2) - (1, 0, 2)) ]

Final answer: ( \vec{r} = (1, 0, 2) + \lambda(3, 3, 0) ).

Example 3

Find the vector equation of the line through ( (0, 1, -1) ) and ( (2, 1, 3) ).

Direction: ( (2, 1, 3) - (0, 1, -1) = (2, 0, 4) )

Using ( \vec{a} = (0, 1, -1) ):

[ \vec{r} = (0, 1, -1) + \lambda(2, 0, 4) ]

Final answer: ( \vec{r} = (0, 1, -1) + \lambda(2, 0, 4) ).

Example 4

Convert ( \vec{r} = (1, 2, 3) + \lambda(2, -1, 4) ) to Cartesian form.

Read off the base point and direction, then build the symmetric form:

[ \frac{x - 1}{2} = \frac{y - 2}{-1} = \frac{z - 3}{4} ]

Final answer: ( \frac{x - 1}{2} = \frac{y - 2}{-1} = \frac{z - 3}{4} ).

Example 5

Find the vector equation of the plane through ( (1, 0, 0) ) with normal ( \vec{N} = (2, 3, 1) ).

Using point-normal form:

[ (\vec{r} - (1, 0, 0)) \cdot (2, 3, 1) = 0 ]

Final answer: ( 2(x - 1) + 3y + z = 0 ), i.e. ( 2x + 3y + z = 2 ).

Example 6

Find the vector equation of the plane through three points A( (1, 0, 0) ), B( (0, 1, 0) ), C( (0, 0, 1) ).

Build two in-plane directions. Their cross product gives the normal:

[ (\vec{b} - \vec{a}) \times (\vec{c} - \vec{a}) = (1, 1, 1) ]

Then:

[ (\vec{r} - \vec{a}) \cdot (1, 1, 1) = 0 ]

Final answer: ( x + y + z = 1 ).

Why Vector Equations Are Worth Learning

Vector equations were the natural language once mathematicians started treating direction as a first-class object.

Where the position-plus-direction idea pays off:

Key Takeaways