To find the velocity of an object at any given time [tex]\( t \)[/tex], we need to determine the derivative of the position vector [tex]\(\vec{r}(t)\)[/tex] with respect to time. The position vector given is:
[tex]\[\vec{r} = t^2 \hat{i} + 2 \hat{j} - t \hat{k}\][/tex]
The velocity [tex]\(\vec{v}(t)\)[/tex] is the time derivative of the position vector [tex]\(\vec{r}(t)\)[/tex]:
[tex]\[\vec{v}(t) = \frac{d\vec{r}}{dt}\][/tex]
Let's differentiate each component of [tex]\(\vec{r}(t)\)[/tex] with respect to time [tex]\( t \)[/tex]:
1. Differentiate [tex]\(t^2\)[/tex] with respect to [tex]\(t\)[/tex]:
[tex]\[
\frac{d(t^2)}{dt} = 2t \quad \text{which gives us the \(\hat{i}\) component of the velocity vector.}
\][/tex]
2. Differentiate [tex]\(2\)[/tex] (a constant) with respect to [tex]\(t\)[/tex]:
[tex]\[
\frac{d(2)}{dt} = 0 \quad \text{which gives us the \(\hat{j}\) component of the velocity vector.}
\][/tex]
3. Differentiate [tex]\(-t\)[/tex] with respect to [tex]\(t\)[/tex]:
[tex]\[
\frac{d(-t)}{dt} = -1 \quad \text{which gives us the \(\hat{k}\) component of the velocity vector.}
\][/tex]
Now, we substitute [tex]\(t = 6\)[/tex] seconds into the differentiated components:
1. The [tex]\(\hat{i}\)[/tex] component:
[tex]\[
2t = 2 \times 6 = 12
\][/tex]
2. The [tex]\(\hat{j}\)[/tex] component:
[tex]\[
0
\][/tex]
3. The [tex]\(\hat{k}\)[/tex] component:
[tex]\[
-1
\][/tex]
Therefore, the velocity vector [tex]\(\vec{v}(t)\)[/tex] at [tex]\(t = 6\)[/tex] seconds is:
[tex]\[
\vec{v}(6) = 12 \hat{i} + 0 \hat{j} - 1 \hat{k}
\][/tex]
Hence, the correct answer among the given choices is:
(4) [tex]\(12 \hat{i} + 2 \hat{j} - \hat{k}\)[/tex]
