example 4 sketch the graph of the following vector function.\n\n$$\\vec{r}(t) = \\langle 2 - 4t, -1 + 5t, 3…

example 4 sketch the graph of the following vector function.\n\n$$\\vec{r}(t) = \\langle 2 - 4t, -1 + 5t, 3 + t \\rangle$$

example 4 sketch the graph of the following vector function.\n\n$$\\vec{r}(t) = \\langle 2 - 4t, -1 + 5t, 3 + t \\rangle$$

Answer

Explanation:

Step1: Identify the vector function components

The vector function is $\vec{r}(t) = \langle 2 - 4t, -1 + 5t, 3 + t \rangle$.

Step2: Separate into point and direction vectors

Rewrite the function in the form $\vec{r}(t) = \vec{r}_0 + t\vec{v}$. $$\vec{r}(t) = \langle 2, -1, 3 \rangle + t\langle -4, 5, 1 \rangle$$

Step3: Identify the geometric shape

The equation represents a straight line in 3D space.

Step4: Determine the starting point and direction

The line passes through $P(2, -1, 3)$ with direction vector $\vec{v} = \langle -4, 5, 1 \rangle$.

Step5: Describe the graph sketch

Plot point $(2, -1, 3)$ and draw a line through it parallel to $\vec{v}$.

Answer:

The graph is a straight line in 3D space passing through the point $(2, -1, 3)$ and extending infinitely in the direction of the vector $\vec{v} = \langle -4, 5, 1 \rangle$. To sketch it, plot the initial point at $t=0$, which is $(2, -1, 3)$, and another point at $t=1$, which is $(-2, 4, 4)$, then connect them with a straight line.