Question

Find the equation of the line through p=(4,2,-5) in the direction of v = (3,16,1).

Answer 1

The equation of the line through the point
`\( P = (4,2,-5) \)` in the direction of
`\( \mathbf{v} = (3,16,1) \)` is given by:

`\[ \mathbf{r}(t) = (4,2,-5) + t(3,16,1) \]`

Step-by-step explanation:

The equation of a line in three-dimensional space is often expressed in vector form as
`\( \mathbf{r}(t) = \mathbf{p} + t\mathbf{v} \)`, where:

-
`\( \mathbf{r}(t) \)` is the position vector of a point on the line,

-
`\( \mathbf{p} \)` is a known point on the line (in this case,
`\( \mathbf{p} = (4,2,-5) \))`,

-
`\( \mathbf{v} \)` is the direction vector of the line (in this case,
`\( \mathbf{v} = (3,16,1) \)`),

-
`\( t \)` is a parameter.

Substituting the given values, the equation of the line is:

`\[ \mathbf{r}(t) = (4,2,-5) + t(3,16,1) \]`

This equation represents the line passing through the point
`\( (4,2,-5) \)` in the direction of
`\( (3,16,1) \).`