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: Where:
- is a point on the line (position vector).
- is a parameter.
- is the direction vector.
In this case, the direction vector is explicitly given as .
2. Vector Equation Form:
A line can also be given in the vector form: Where:
- is a fixed point on the line (position vector).
- is the direction vector. So, in this case, is already the direction vector.
3. Symmetric Form:
The symmetric form of a line in 3D is: The direction vector can be directly extracted as , where , , and are the coefficients in the denominator of the symmetric equations.
4. Cartesian Form (Two-Point Form):
If a line is defined by two points, and , the direction vector is found by subtracting the coordinates of these points:
Example:
For the symmetric form of the line: The direction vector is .
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:
- How do you find the direction vector from a point and a line equation?
- How can the direction vector be normalized?
- How do you convert a Cartesian equation to parametric form?
- What is the significance of a direction vector in vector calculus?
- Can the direction vector determine if two lines are parallel?
Tip: The magnitude of a direction vector can be computed using the Pythagorean theorem.
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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
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