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: $\vec{r}(t) = \vec{r}_0 + t\vec{d}$ Where:

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

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

2. Vector Equation Form:

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

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

3. Symmetric Form:

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

4. Cartesian Form (Two-Point Form):

If a line is defined by two points, $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, the direction vector is found by subtracting the coordinates of these points: $\text{Direction vector} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle$

Example:

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

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.