Use vector equations of curves in two or three dimensions involving a parameter, and determine a ‘corresponding’ Cartesian equation in the two-dimensional case.

A curve in two dimensions is given by the vector equation \(\mathbf{r}(t) = (3\cos t)\mathbf{i} + (4\sin t)\mathbf{j}\) for \(0 \leq t \leq 2\pi\). (a) Express \(\cos t\) and \(\sin t\) separately in terms of \(x\) and \(y\). (1 mark) (b) Use your results from part (a) to determine the corresponding Cartesian equation of the curve. (1 mark) (c) Identify the geometric shape traced by this curve. (1 mark)

A particle moves along a path described by the vector equation \(\vec{r} = (3-t)\mathbf{i} + (t^2-2t)\mathbf{j}\) for \(t \geq 0\). Determine the corresponding Cartesian equation of the path.

A curve is defined by the vector equation \(\vec{r}(t) = (3t - 1)\mathbf{i} + (2t^2 + 5)\mathbf{j}\) for \(t \in \mathbb{R}\). (a) Express \(t\) in terms of \(x\). (1 mark) (b) Hence determine the corresponding Cartesian equation in the form \(y = f(x)\). (1 mark)

A curve in two dimensions is defined by the vector equation \(\mathbf{r} = (3t - 1)\mathbf{i} + (2t^2 + 5)\mathbf{j}\) for \(t \in \mathbb{R}\). (a) Express \(t\) in terms of \(x\). (1 mark) (b) Hence determine the corresponding Cartesian equation in the form \(y = f(x)\). (2 marks)

A curve is defined by the vector equation \(\mathbf{r} = (3t - 1)\mathbf{i} + (2t^2 + 1)\mathbf{j}\) for \(t \geq 0\). Which of the following is the corresponding Cartesian equation?