Converting the Vector Equation of a Line into Cartesian Form

Converting the Vector Equation of a Line into Cartesian Form

A straight line in three-dimensional space can be expressed using vectors. One important vector form is

(𝐑 − 𝐀) × 𝐁 = 0

This equation states that the displacement vector from a fixed point 𝐀 to a general point 𝐑 is parallel to the direction vector 𝐁. Two non-zero vectors have a zero cross product precisely when they are parallel. From this fact, the Cartesian (symmetric) equation of the line can be derived.

Converting the Vector Equation of a Line into Cartesian Form


1. Substituting Coordinate Vectors

The general point on the line is written as

𝐑 = (x, y, z)

The fixed point is

𝐀 = (x₁, y₁, z₁)

The direction vector is

𝐁 = (l, m, n)

Substituting these into the vector equation yields:

((x, y, z) − (x₁, y₁, z₁)) × (l, m, n) = 0

which simplifies to:

(x − x₁, y − y₁, z − z₁) × (l, m, n) = 0

2. Using the Condition for a Zero Cross Product

If two non-zero vectors have a zero cross product, then one is a scalar multiple of the other. Therefore, there exists a scalar λ such that

(x − x₁, y − y₁, z − z₁) = λ(l, m, n)

This produces the system:

  • x − x₁ = λl
  • y − y₁ = λm
  • z − z₁ = λn

3. Cartesian (Symmetric) Form of the Line

Each of the three equations expresses the same scalar λ. Eliminating λ gives the symmetric Cartesian form:

(x − x₁)/l = (y − y₁)/m = (z − z₁)/n

This expresses the fact that the displacement from the fixed point 𝐀 to any point on the line is proportional to the components of the direction vector 𝐁.

Conclusion

Starting from the vector equation (𝐑 − 𝐀) × 𝐁 = 0, the Cartesian equation of a line in 3D is obtained by substituting coordinate vectors and applying the condition for a zero cross product. The resulting form

(x − x₁)/l = (y − y₁)/m = (z − z₁)/n

makes both the direction ratios and the reference point of the line explicitly visible, providing a clear and convenient representation in coordinate geometry.

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