Geometric interpretation of systems

10 Three planes have equations $$\begin{aligned} - x + 2 y + z & = 0 \\ 2 x - y - z & = 0 \\ x + y & = a \end{aligned}$$ where \(a\) is a constant.

1. Investigate the arrangement of the planes:
   • when \(a = 0\);
   • when \(a \neq 0\).
   • Chris claims that the position vectors \(- \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , 2 \mathbf { i } - \mathbf { j } - \mathbf { k }\) and \(\mathbf { i } + \mathbf { j }\) lie in a plane. Determine whether or not Chris is correct.

8 The equations of three planes are $$\begin{array} { r } 2 x + y + 3 z = 3 \\ 3 x - y - 2 z = 2 \\ - 4 x + 3 y + 7 z = k \end{array}$$ where \(k\) is a constant.

1. By considering a suitable determinant, show that the planes do not meet at a single point.
2. Given that the planes form a sheaf, determine the value of \(k\).

7 Three planes have equations $$\begin{array} { r } x + 2 y - 3 z = 0 \\ - x + 3 y - 2 z = 0 \\ x - 2 y + k z = k \end{array}$$ where \(k\) is a constant.

1. For the case \(k = 0\), the origin lies on all three planes. Use a determinant to explain whether there are any other points that lie on all three planes in this case.
2. You are now given that \(k = 1\).
   1. Show that there are no points that lie on all three planes.
   2. Describe the geometrical arrangement of the three planes.

9 Three planes have equations \(k x + y - 2 z = 0\) \(2 x + 3 y - 6 z = - 5\) \(3 x - 2 y + 5 z = 1\) where \(k\) is a constant. Investigate the arrangement of the planes for each of the following cases. If in either case the planes meet at a unique point, find the coordinates of that point.

1. \(k = - 1\)
2. \(k = \frac { 2 } { 3 }\)

13 Matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c c } k & 1 & - 5 \\ 2 & 3 & - 3 \\ - 1 & 2 & 2 \end{array} \right)\), where \(k\) is a constant.

1. Show that \(\operatorname { det } \mathbf { M } = 12 ( k - 3 )\).
2. Find a solution of the following simultaneous equations for which \(x \neq z\). $$\begin{aligned} 4 x ^ { 2 } + y ^ { 2 } - 5 z ^ { 2 } & = 6 \\ 2 x ^ { 2 } + 3 y ^ { 2 } - 3 z ^ { 2 } & = 6 \\ - x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } & = - 6 \end{aligned}$$
3. (A) Verify that the point ( \(2,0,1\) ) lies on each of the following three planes. $$\begin{aligned} 3 x + y - 5 z & = 1 \\ 2 x + 3 y - 3 z & = 1 \\ - x + 2 y + 2 z & = 0 \end{aligned}$$ (B) Describe how the three planes in part (iii) (A) are arranged in 3-D space. Give reasons for your answer.
4. Find the values of \(k\) for which the transformation represented by \(\mathbf { M }\) has a volume scale factor of 6 .

7. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 3 & k & 4 \\ 3 & 2 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Find the values of \(k\) for which the matrix \(\mathbf { M }\) has an inverse.
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect $$\begin{aligned} & 2 x - y + z = p \\ & 3 x - 6 y + 4 z = 1 \\ & 3 x + 2 y - z = 0 \end{aligned}$$
3. 1. Find the value of \(q\) for which the set of simultaneous equations $$\begin{aligned} & 2 x - y + z = 1 \\ & 3 x - 5 y + 4 z = q \\ & 3 x + 2 y - z = 0 \end{aligned}$$ can be solved.
   2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically.

6. $$\mathbf { M } = \left( \begin{array} { r r r } k & 5 & 7 \\ 1 & 1 & 1 \\ 2 & 1 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Given that \(k \neq 4\), find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\).
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 2 x + 5 y + 7 z = 1 \\ x + y + z = p \\ 2 x + y - z = 2 \end{array}$$
3. 1. Find the value of \(q\) for which the following planes intersect in a straight line. $$\begin{array} { r } 4 x + 5 y + 7 z = 1 \\ x + y + z = q \\ 2 x + y - z = 2 \end{array}$$
   2. For this value of \(q\), determine a vector equation for the line of intersection.

12 Three planes have equations $$\begin{aligned} 4 x - 5 y + z & = 8 \\ 3 x + 2 y - k z & = 6 \\ ( k - 2 ) x + k y - 8 z & = 6 \end{aligned}$$ where \(k\) is a real constant. The planes do not meet at a unique point. 12

1. Find the possible values of \(k\).
   12
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system.

7 Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3 \\ ( k - 1 ) x + ( 3 - k ) y + 2 z & = 1 \\ 7 x - 3 y + 4 z & = 2 \end{aligned}$$ 7

1. The planes do not meet at a unique point.
   Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
   

   7

   For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)