To interpret your question, you're describing a line in vector form, with a direction vector and a point on the line.

The general equation for a line in vector form is:

$\mathbf{r} = \mathbf{r_0} + t \mathbf{d}$

Where:

• $\mathbf{r}$ is the position vector of any point on the line.
• $\mathbf{r_0}$ is a fixed point on the line (often referred to as the position vector of a point on the line).
• $t$ is a scalar parameter.
• $\mathbf{d}$ is the direction vector, which defines the direction of the line.

Let's break down your expression step by step:

1. Position vector of a point on the line: You have the expression: $\mathbf{r}^ = 2\hat{j} - 3\hat{k} - \hat{i} + \text{neu}$ Assuming "neu" is a typo or a misrepresentation of a constant value. If "neu" is meant to represent something like a scalar multiple or another vector, please clarify.

2. Direction vector: You also mention that the line is along the direction vector: $\hat{j} - 3\hat{k} + 5\hat{i}$ So, the direction vector $\mathbf{d}$ is: $\mathbf{d} = \hat{i} + \hat{j} - 3\hat{k}$

Possible Corrected Equation:

If I assume "neu" is irrelevant or a mistake and you meant a simple line equation with a point and direction, the equation of the line would look like this:

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

Where:

• $\mathbf{r_0} = -\hat{i} + 2\hat{j} - 3\hat{k}$ is the point on the line.
• $\mathbf{d} = \hat{i} + \hat{j} - 3\hat{k}$ is the direction vector.

Clarifications:

1. Are you asking to find a specific point on the line, or do you need the parametric equations for this line?
2. Is "neu" meant to be part of the position vector or a scalar factor?
3. Would you like to know how to convert this line into parametric form, with separate equations for each coordinate?

Let me know if you want me to go deeper into any of these aspects!

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Related Questions:

1. How can you write the parametric equations for a line in 3D?
2. What does the direction vector of a line represent geometrically?
3. How do you find the equation of a line given a point and a direction vector?
4. How do you interpret a vector equation like $\mathbf{r}(t) = \mathbf{r_0} + t\mathbf{d}$?
5. What happens if the direction vector is zero in a line equation?

Tip: Remember that the direction vector defines how the line "moves" from the point. If you change the direction vector, the line will rotate or shift in space!