Answer
Velocity: \[ \mathbf{v}(t) = (1 - \cos(t))\mathbf{i} + \sin(t)\mathbf{j} + e^{t}\mathbf{k} \] Position: \[ \mathbf{r}(t) = (t - \sin(t) - 1)\mathbf{i} + (1 - \cos(t))\mathbf{j} + e^{t}\mathbf{k} \]
Work Step by Step
Step 1 Given: \[ \mathbf{a}(t) = \sin(t)\mathbf{i} + \cos(t)\mathbf{j} + e^{t}\mathbf{k} \] with initial conditions: \[ \mathbf{v}(0) = \mathbf{k}, \quad \mathbf{r}(0) = -\mathbf{i} + \mathbf{k} \] Step 2 To find the velocity: \[ \mathbf{v}(t) = \int \mathbf{a}(t) \, dt = \int (\sin(t)\mathbf{i} + \cos(t)\mathbf{j} + e^{t}\mathbf{k}) \, dt \] \[ = (-\cos(t)\mathbf{i} + \sin(t)\mathbf{j} + e^{t}\mathbf{k}) + \mathbf{c} \] Since \(\mathbf{v}(0) = \mathbf{k}\), then: \[ \mathbf{c} = \mathbf{k} \] So, \[ \mathbf{v}(t) = (1 - \cos(t))\mathbf{i} + \sin(t)\mathbf{j} + e^{t}\mathbf{k} \] Step 3 To find the position: \[ \mathbf{r}(t) = \int \mathbf{v}(t) \, dt = \int ((1 - \cos(t))\mathbf{i} + \sin(t)\mathbf{j} + e^{t}\mathbf{k}) \, dt \] \[ = ((t - \sin(t))\mathbf{i} - \cos(t)\mathbf{j} + e^{t}\mathbf{k}) + \mathbf{c}_1 \] Since \(\mathbf{r}(0) = -\mathbf{i} + \mathbf{k}\), then: \[ \mathbf{c}_1 = -\mathbf{i} + \mathbf{k} \] So, \[ \mathbf{r}(t) = (t - \sin(t) - 1)\mathbf{i} + (1 - \cos(t))\mathbf{j} + e^{t}\mathbf{k} \] Result Velocity: \[ \mathbf{v}(t) = (1 - \cos(t))\mathbf{i} + \sin(t)\mathbf{j} + e^{t}\mathbf{k} \] Position: \[ \mathbf{r}(t) = (t - \sin(t) - 1)\mathbf{i} + (1 - \cos(t))\mathbf{j} + e^{t}\mathbf{k} \]
