Define and use the vector (cross) product to determine a vector normal to a given plane, with and without technology. 𝒂 × 𝒃 = |𝒂| |𝒃| sin(𝜃) 𝒏̂ 𝒂 × 𝒃 = ( 𝑎1 𝑎2 𝑎3) × ( 𝑏1 𝑏2 𝑏3) = ( 𝑎2𝑏3 − 𝑎3𝑏2 𝑎3𝑏1 − 𝑎1𝑏3 𝑎1𝑏2 − 𝑎2𝑏1)

Consider the points \(P(2, 1, -1)\), \(Q(4, 0, 3)\) and \(R(3, -2, 1)\) which lie on the plane \(\pi\). a) Determine two vectors that lie in the plane \(\pi\). (1 mark) b) Use the vector product to find a vector \(\mathbf{n}\) that is normal to the plane \(\pi\). (2 marks) c) Hence determine the Cartesian equation of the plane \(\pi\). (1 mark)

Three points \(P\), \(Q\) and \(R\) lie on a plane and have position vectors \(\mathbf{p} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix}\), \(\mathbf{q} = \begin{pmatrix} 3 \\ -1 \\ 5 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\). (a) Determine two direction vectors that lie in the plane. (b) Use the vector product to determine a vector \(\mathbf{n}\) that is normal to the plane. (c) Hence, find a unit vector normal to the plane.

The table below shows the position vectors of four points P, Q, R and S. (a) Determine two vectors that lie in the plane containing P, Q and R. (1 mark) (b) Determine a vector \(\mathbf{m}\) that is normal to the plane containing P, Q and R. (1 mark) (c) Calculate \(|\mathbf{m}|\). (1 mark) (d) Verify, using a vector product method, that the point S does not lie on the plane containing P, Q and R. (1 mark)

A plane passes through the origin and contains the vectors \(\mathbf{u} = [2, 1, -1]\) and \(\mathbf{v} = [1, -2, 3]\). Which vector is normal to this plane?

Consider the four points P(2, -1, 3), Q(4, 0, 1), R(3, 2, -1), and S(1, k, 2), where k is a real constant. (a) Determine two non-parallel vectors that lie in the plane containing P, Q, and R. (1 mark) (b) Use a vector product method, without technology, to determine a unit vector n̂ that is normal to the plane containing P, Q, and R. (2 marks) (c) Hence determine the value of k such that S also lies in the plane containing P, Q, and R. (2 marks)

A triangular region in three-dimensional space has vertices at \(P(2, -1, 3)\), \(Q(4, 2, 1)\), and \(R(3, 0, 5)\). (a) Determine two vectors that lie in the plane containing \(P\), \(Q\), and \(R\). (1 mark) (b) Use the vector product to determine a vector that is perpendicular to the plane. (2 marks) (c) Hence, determine a unit vector normal to the plane. (1 mark) (d) Verify your result by showing that the unit vector is perpendicular to both vectors found in part (a). (1 mark)

Consider the plane containing the points \(P(2,\,1,\,3)\), \(Q(4,\,0,\,2)\) and \(R(3,\,2,\,1)\). (a) Determine two vectors that lie in the plane. (1 mark) (b) Use a vector product method to determine a vector \(\mathbf{n}\) that is normal to the plane. (1 mark)

Consider the three points \(P(2, -1, 4)\), \(Q(5, 0, 3)\) and \(R(4, 2, 1)\) which define a plane \(\pi\). (a) Determine two direction vectors in the plane \(\pi\). (1 mark) (b) Use the vector product to determine a non-zero vector \(\mathbf{n}\) that is perpendicular to the plane \(\pi\), showing all working without technology. (2 marks) (c) Hence determine a unit vector \(\hat{\mathbf{n}}\) that is normal to the plane \(\pi\), expressing your answer with components correct to three decimal places. (2 marks)

A telecommunications tower is anchored by three support cables attached at points \(P\), \(Q\), and \(R\). In a coordinate system with the tower base at the origin (units in metres), the attachment points are \(P(2, 1, 4)\), \(Q(5, 3, 4)\), and \(R(2, 4, 4)\). (a) Determine two vectors that lie in the plane containing \(P\), \(Q\), and \(R\). (b) Using a vector product method, determine a vector \(\mathbf{m}\) that is perpendicular to the plane. (c) Hence, determine a unit vector normal to the plane.

A plane contains the vectors \(\begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\) and \(\begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix}\). Which one of the following vectors is normal to the plane?

An aircraft control surface is triangular with vertices at points P, Q and R, where P = (2, 1, 4), Q = (3, 4, 5) and R = (4, 2, 7). The surface is shown in the stimulus. Determine a unit vector **u** that is normal to the plane containing the control surface.

Consider the plane that contains the point \(P(2, -1, 3)\) and the vectors \(\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix}\). (a) Determine a vector \(\mathbf{n}\) that is normal to the plane. (1 mark) (b) Hence determine the scalar equation of the plane in the form \(ax + by + cz = d\). (1 mark)