Determine a vector between two points.

In a three-dimensional Cartesian coordinate system, points \(P\), \(Q\), \(R\) and \(S\) have position vectors \(\vec{OP} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}\), \(\vec{OQ} = \begin{pmatrix} 5 \\ 3 \\ 1 \end{pmatrix}\), \(\vec{OR} = \begin{pmatrix} -3 \\ 2 \\ 0 \end{pmatrix}\) and \(\vec{OS} = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix}\). (a) Determine the vector \(\vec{PQ}\). (1 mark) (b) Determine the vector \(\vec{RS}\). (1 mark) (c) Point \(T\) divides \(\vec{PQ}\) in the ratio \(2:1\) from \(P\). Determine the position vector \(\vec{OT}\). (1 mark)

In a three-dimensional Cartesian coordinate system, four points P, Q, R, and S are defined such that P(2, -1, 4), Q(5, 3, 1), R(-3, 2, 5), and S(k, 7, -2), where k ∈ ℝ. Use the diagram showing the configuration of these points in 3D space to answer the following questions. (a) Determine the vector PQ. (1 mark) (b) Point T divides QR in the ratio 3:2 from Q to R. Determine the vector PT. (2 marks) (c) Given that PT and PS are perpendicular, determine the value of k. (2 marks)

A 3D Cartesian coordinate system has points \(P(2, -1, 4)\), \(Q(-3, 5, 1)\), and \(R(7, 0, -2)\). (a) Determine the vector \(\overrightarrow{PQ}\). (1 mark) (b) Point \(S\) divides \(PR\) in the ratio \(3:2\) from \(P\). Determine the position vector \(\overrightarrow{OS}\), where \(O\) is the origin. (2 marks) (c) Hence determine the vector \(\overrightarrow{QS}\). (1 mark)

Let \(P\) and \(Q\) be points in three-dimensional space with coordinates \(P(4, -1, 5)\) and \(Q(2, 3, -2)\). Determine the vector \(\overrightarrow{PQ}\).

In three-dimensional space, point \(P\) has coordinates \((4, -2, 5)\) and point \(Q\) has coordinates \((-1, 3, 2)\). (a) Determine the vector \(\overrightarrow{PQ}\). (1 mark) (b) Determine the magnitude of \(\overrightarrow{PQ}\), correct to two decimal places. (1 mark) (c) Point \(R\) lies on the line segment \(PQ\) such that \(PR:RQ = 2:3\). Determine the coordinates of \(R\). (1 mark)

In a three-dimensional coordinate system, points \(P\), \(Q\), and \(R\) have coordinates \(P(2, -1, 4)\), \(Q(5, 3, 1)\), and \(R(-3, 2, 7)\). Point \(T\) divides the line segment \(PR\) in the ratio \(2:3\) from \(P\). (a) Determine the vector \(\overrightarrow{PT}\). (1 mark) (b) Determine the position vector \(\overrightarrow{OT}\). (1 mark) (c) Hence determine the vector \(\overrightarrow{TQ}\). (1 mark)

In three-dimensional space, point \(P\) has coordinates \((4, -2, 7)\) and point \(Q\) has coordinates \((-1, 3, 2)\). Determine the vector \(\overrightarrow{PQ}\).

Determine the vector \(\overrightarrow{PQ}\) where \(P\) has coordinates \((2, -3, 5)\) and \(Q\) has coordinates \((7, 1, -2)\).