Determine vector, parametric and Cartesian equations of straight lines and straight-line segments given the position of two points, or equivalent information, in both two and three dimensions. vector equation of line: 𝒓 = 𝒂 + 𝑡𝒅 parametric equations of line: 𝑥 = 𝑎1 + 𝑡 𝑑1 𝑦 = 𝑎2 + 𝑡 𝑑2 𝑧 = 𝑎3 + 𝑡 𝑑3 Cartesian equation of line: 𝑥−𝑎1 𝑑1 = 𝑦−𝑎2 𝑑2 = 𝑧−𝑎3 𝑑3

The table below shows coordinates of three points \(P\), \(Q\), and \(R\) that lie on three different lines in three-dimensional space. Line \(L_1\) passes through points \(P\) and \(Q\). Line \(L_2\) passes through point \(R\) and is parallel to the vector \(\vec{v} = 3\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}\). Determine: (a) the vector equation of line \(L_1\) in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{d}\) (2 marks) (b) the parametric equations of line \(L_2\) (1 mark) (c) the Cartesian equation of line \(L_1\) (2 marks)

The table below shows the coordinates of three points \(P\), \(Q\), and \(R\) in three-dimensional space. Determine: (a) the coordinates of the midpoint \(M\) of \(PQ\) (b) the direction vector from \(M\) to \(R\) (c) the Cartesian equation of the line that passes through \(M\) and \(R\).

Consider the line \(L\) that passes through point \(P(2, -3, 5)\) and is parallel to the vector \(\vec{d} = 4\mathbf{i} - 2\mathbf{j} + 6\mathbf{k}\). a) Determine the vector equation of line \(L\). (1 mark) b) Determine the parametric equations of line \(L\). (1 mark) c) Determine the Cartesian equation of line \(L\). (1 mark)

A line L passes through the point P(4, -3, 7) and the point Q(1, 3, 5). (a) Determine the Cartesian equation of line L. (2 marks) (b) Determine the parametric equations of line L. (2 marks) (c) Determine the coordinates of the point where L intersects the xy-plane. (1 mark)

A line L passes through point A(2, -1, 3) and point B(5, 3, -1). (a) Determine the vector equation of line L in the form r = a + td. (1 mark) (b) Determine the parametric equations of line L. (1 mark) (c) Express the equation of line L in Cartesian form. (2 marks) (d) Determine the coordinates of point C on line L where t = 3. (1 mark)

Consider points \(M(4, -2, 1)\) and \(N(-2, 4, 7)\). a) Determine the direction vector of the line passing through \(M\) and \(N\). (1 mark) b) Hence determine the vector equation of the line passing through \(M\) and \(N\). Express your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{d}\). (1 mark) c) Express the equation of this line in Cartesian form. (1 mark)

A line passes through the points \(P(1, -2, 4)\) and \(Q(3, 1, -2)\). Which of the following is the Cartesian equation of this line?

A telecommunications cable runs in a straight line from transmission tower P at coordinates (2, -3, 5) through relay station Q at coordinates (4, 1, 2). All coordinates are measured in kilometres, where the xy-plane represents ground level and the z-axis measures height above ground. (a) Determine the direction vector of the cable. (1 mark) (b) Express the cable's path as a vector equation. (1 mark) (c) Determine the Cartesian equation of the line containing the cable. (1 mark)

Consider the points \(M(4, -2, 1)\) and \(N(-2, 4, 7)\) in three-dimensional space. a) Determine the vector equation of the line passing through \(M\) and \(N\). (1 mark) b) Determine the parametric equations of this line. (1 mark) c) Express the line in Cartesian form. (1 mark)

A telecommunications tower is modelled in a three-dimensional coordinate system where distances are measured in metres. The base of a signal cable is located at point \(P(2, -3, 1)\) and the cable is attached to an anchor point \(Q(6, 1, -5)\). (a) Determine the direction vector of the cable. [1 mark] (b) Determine the vector equation of the line representing the cable, expressing your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{d}\). [1 mark] (c) Determine the Cartesian equation of the line representing the cable. [1 mark]