Use vector methods in applications, including areas of shapes and determining vector and Cartesian equations of a plane and of regions in a plane. vector equation of plane: 𝒓 ⋅ 𝒏 = 𝒂 ⋅ 𝒏 Cartesian equation of plane: 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0

Use the given vector equation \(\mathbf{r} \cdot \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} = 7\) to determine the Cartesian equation of the plane.

A plane φ passes through the three points P, Q, and R, whose coordinates are given in the table below. *Refer to the table below.* (a) Determine two non-parallel vectors that lie in the plane φ. (1 mark) (b) Determine a vector perpendicular to the plane φ. (1 mark) (c) Determine a unit vector perpendicular to the plane φ. (1 mark) (d) Determine the Cartesian equation of the plane φ in the form ax + by + cz + d = 0, where a, b, c, and d are integers with no common factor. (2 marks)

Three control points on a sloping plane surface are recorded as \(P(4, 1, 2)\), \(Q(7, 3, 5)\), and \(R(5, 4, 3)\). (a) Determine two direction vectors that lie in the plane. (b) Calculate the normal vector to the plane using the cross product. (c) Determine the Cartesian equation of the plane in the form \(ax + by + cz + d = 0\).

A plane passes through the point \(P(4, -2, 1)\) and has normal vector \(\mathbf{n} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}\). Use the given information to determine the Cartesian equation of the plane.