1. The problem is to express the vector function \( \mathbf{r}(x) = (-2x^3 + x^2 - 1)e^{-2x+1} \) in terms of the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) with the condition that \( y = x \). 2. Since \( y = x \), we can write the vector function as \( \mathbf{r}(x) = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \). Here, \( y = x \), so \( \mathbf{r}(x) = x \mathbf{i} + x \mathbf{j} + z \mathbf{k} \). 3. The given scalar function is \( f(x) = (-2x^3 + x^2 - 1)e^{-2x+1} \). We can assign this to the \( z \)-component, so \( z = f(x) \). 4. Therefore, the vector function in \( (\mathbf{i}, \mathbf{j}, \mathbf{k}) \) form is: $$ \mathbf{r}(x) = x \mathbf{i} + x \mathbf{j} + (-2x^3 + x^2 - 1)e^{-2x+1} \mathbf{k} $$ This expresses the vector in terms of \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) with \( y = x \). Final answer: $$ \mathbf{r}(x) = x \mathbf{i} + x \mathbf{j} + (-2x^3 + x^2 - 1)e^{-2x+1} \mathbf{k} $$