Identify the direction vector

Oct 13, 2025 by windailys.com 51 views

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Introduction

In vector calculus, the vector equation of a line is a way to represent a line in three-dimensional space using vectors. It is a powerful tool for solving problems in geometry, physics, and engineering. In this article, we will discuss how to find the vector equation of a line passing through a given point and parallel to a vector.

What is the Vector Equation of a Line?

The vector equation of a line is given by:

\mathbf{r} = \mathbf{r}_0 + t \mathbf{d}

where:

$\mathbf{r}$ is the position vector of a point on the line
$\mathbf{r}_0$ is the position vector of a fixed point on the line (called the initial point)
$\mathbf{d}$ is the direction vector of the line
$t$ is a parameter that varies along the line

Finding the Vector Equation of a Line Passing Through a Given Point and Parallel to a Vector

To find the vector equation of a line passing through a given point and parallel to a vector, we need to follow these steps:

Identify the given point: The given point is represented by the position vector $\mathbf{r}_0 = 2 \mathbf{i} + 3 \mathbf{j} + \mathbf{k}$ .
Identify the direction vector: The direction vector $\mathbf{d}$ is parallel to the vector $4 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k}$ . Since the direction vector is parallel to the given vector, we can use the given vector as the direction vector.
Write the vector equation of the line: Using the formula $\mathbf{r} = \mathbf{r}_0 + t \mathbf{d}$ , we can write the vector equation of the line as:

\mathbf{r} = (2 \mathbf{i} + 3 \mathbf{j} + \mathbf{k}) + t (4 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k})

Simplifying the Vector Equation of the Line

We can simplify the vector equation of the line by combining like terms:

\mathbf{r} = (2 + 4t) \mathbf{i} + (3 - 2t) \mathbf{j} + (1 + 3t) \mathbf{k}

Conclusion

In this article, we discussed how to find the vector equation of a line passing through a given point and parallel to a vector. We identified the given point, the direction vector, and wrote the vector equation of the line using the formula $\mathbf{r} = \mathbf{r}_0 + t \mathbf{d}$ . We also simplified the vector equation of the line by combining like terms.

Example Use Cases

The vector equation of a line has many practical applications in physics, engineering, and computer science. Here are a few example use cases:

Motion of an object: The vector equation of a line can be used to describe the motion of an object in three-dimensional space.
Trajectory of a projectile: The vector equation of a line can be used to describe the trajectory of a projectile under the influence of gravity.
Computer graphics: The vector equation of a line can be used to create 3D models and animations in computer graphics.

Final Thoughts

The vector equation of a line is a powerful tool for solving problems in geometry, physics, and engineering. By following the steps outlined in this article, you can find the vector equation of a line passing through a given point and parallel to a vector. With practice and experience, you can apply the vector equation of a line to a wide range of problems and applications.

References

[1] Vector Calculus by Michael Corral
[2] Linear Algebra and Its Applications by Gilbert Strang
[3] Calculus: Early Transcendentals by James Stewart

Q: What is the vector equation of a line?

A: The vector equation of a line is a way to represent a line in three-dimensional space using vectors. It is given by the formula:

\mathbf{r} = \mathbf{r}_0 + t \mathbf{d}

where:

$\mathbf{r}$ is the position vector of a point on the line
$\mathbf{r}_0$ is the position vector of a fixed point on the line (called the initial point)
$\mathbf{d}$ is the direction vector of the line
$t$ is a parameter that varies along the line

Q: How do I find the vector equation of a line passing through a given point and parallel to a vector?

A: To find the vector equation of a line passing through a given point and parallel to a vector, you need to follow these steps:

Identify the given point: The given point is represented by the position vector $\mathbf{r}_0$ .
Identify the direction vector: The direction vector $\mathbf{d}$ is parallel to the given vector. Since the direction vector is parallel to the given vector, you can use the given vector as the direction vector.
Write the vector equation of the line: Using the formula $\mathbf{r} = \mathbf{r}_0 + t \mathbf{d}$ , you can write the vector equation of the line.

Q: What is the difference between the vector equation of a line and the parametric equation of a line?

A: The vector equation of a line and the parametric equation of a line are two different ways to represent a line in three-dimensional space. The vector equation of a line uses vectors to represent the line, while the parametric equation of a line uses coordinates to represent the line.

Q: How do I find the parametric equation of a line from the vector equation of a line?

A: To find the parametric equation of a line from the vector equation of a line, you need to follow these steps:

Write the vector equation of the line: The vector equation of the line is given by $\mathbf{r} = \mathbf{r}_0 + t \mathbf{d}$ .
Extract the coordinates: From the vector equation of the line, you can extract the coordinates of the line as:

x = x_0 + t d_x

y = y_0 + t d_y

z = z_0 + t d_z

where:

$(x_0, y_0, z_0)$ is the initial point of the line
$(d_x, d_y, d_z)$ is the direction vector of the line

Q: What is the significance of the parameter $t$ in the vector equation of a line?

A: The parameter $t$ in the vector equation of a line represents the distance along the line from the initial point. As $t$ varies, the point on the line moves along the line.

Q: How do I find the vector equation of a line passing through two points?

A: To find the vector equation of a line passing through two points, you need to follow these steps:

Identify the two points: The two points are represented by the position vectors $\mathbf{r}_0$ and $\mathbf{r}_1$ .
Find the direction vector: The direction vector $\mathbf{d}$ is given by the difference between the two position vectors:

\mathbf{d} = \mathbf{r}_1 - \mathbf{r}_0

Write the vector equation of the line: Using the formula $\mathbf{r} = \mathbf{r}_0 + t \mathbf{d}$ , you can write the vector equation of the line.

Q: What is the relationship between the vector equation of a line and the Cartesian equation of a line?

A: The vector equation of a line and the Cartesian equation of a line are two different ways to represent a line in three-dimensional space. The vector equation of a line uses vectors to represent the line, while the Cartesian equation of a line uses coordinates to represent the line.

Q: How do I find the Cartesian equation of a line from the vector equation of a line?

A: To find the Cartesian equation of a line from the vector equation of a line, you need to follow these steps:

Write the vector equation of the line: The vector equation of the line is given by $\mathbf{r} = \mathbf{r}_0 + t \mathbf{d}$ .
Extract the coordinates: From the vector equation of the line, you can extract the coordinates of the line as:

x = x_0 + t d_x

y = y_0 + t d_y

z = z_0 + t d_z

where:

$(x_0, y_0, z_0)$ is the initial point of the line
$(d_x, d_y, d_z)$ is the direction vector of the line

Q: What is the significance of the Cartesian equation of a line?

A: The Cartesian equation of a line is a way to represent a line in three-dimensional space using coordinates. It is a powerful tool for solving problems in geometry, physics, and engineering.

References

[1] Vector Calculus by Michael Corral
[2] Linear Algebra and Its Applications by Gilbert Strang
[3] Calculus: Early Transcendentals by James Stewart