Math Problem Statement

how to find direction vector from equation of a line

Solution

To find the direction vector from the equation of a line, it depends on the form in which the line equation is given. Let's cover two common forms of line equations:

1. Parametric Form:

In the parametric form, a line is given as: r(t)=r0+td\vec{r}(t) = \vec{r}_0 + t\vec{d} Where:

  • r0\vec{r}_0 is a point on the line (position vector).
  • tt is a parameter.
  • d\vec{d} is the direction vector.

In this case, the direction vector is explicitly given as d\vec{d}.

2. Vector Equation Form:

A line can also be given in the vector form: r=a+tb\vec{r} = \vec{a} + t\vec{b} Where:

  • a\vec{a} is a fixed point on the line (position vector).
  • b\vec{b} is the direction vector. So, in this case, b\vec{b} is already the direction vector.

3. Symmetric Form:

The symmetric form of a line in 3D is: xx1a=yy1b=zz1c\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} The direction vector can be directly extracted as a,b,c\langle a, b, c \rangle, where aa, bb, and cc are the coefficients in the denominator of the symmetric equations.

4. Cartesian Form (Two-Point Form):

If a line is defined by two points, (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z2)(x_2, y_2, z_2), the direction vector is found by subtracting the coordinates of these points: Direction vector=x2x1,y2y1,z2z1\text{Direction vector} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle

Example:

For the symmetric form of the line: x12=y+34=z56\frac{x - 1}{2} = \frac{y + 3}{4} = \frac{z - 5}{6} The direction vector is 2,4,6\langle 2, 4, 6 \rangle.

Summary:

  • Parametric/vector form: The direction vector is given directly.
  • Symmetric form: Extract the coefficients in the denominator.
  • From two points: Subtract one point from the other.

Would you like more details on any of these forms?

5 Related Questions:

  1. How do you find the direction vector from a point and a line equation?
  2. How can the direction vector be normalized?
  3. How do you convert a Cartesian equation to parametric form?
  4. What is the significance of a direction vector in vector calculus?
  5. Can the direction vector determine if two lines are parallel?

Tip: The magnitude of a direction vector can be computed using the Pythagorean theorem.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Vector Algebra
Line Equations
Direction Vectors
Symmetric Form of a Line
Parametric Equations

Formulas

Symmetric form of a line: (x - x1)/a = (y - y1)/b = (z - z1)/c
Direction vector from two points: ⟨x2 - x1, y2 - y1, z2 - z1⟩
Parametric form of a line: r(t) = r0 + t * d

Theorems

Vector addition
Properties of direction vectors in space

Suitable Grade Level

Grades 10-12