Line of intersection of planes

The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) + \mu ( - \mathbf { i } + \mathbf { k } )\). Obtain a cartesian equation of \(\Pi _ { 1 }\) in the form \(p x + q y + r z = d\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) = 12\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + ( 2 a + 1 ) \mathbf { j } - 3 \mathbf { k }\) and is parallel to \(3 c \mathbf { i } - 3 \mathbf { j } + c \mathbf { k }\), where \(a\) and \(c\) are positive constants. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\frac { 15 } { \sqrt { } 6 }\) and that the acute angle between \(l\) and \(\Pi _ { 1 }\) is \(\sin ^ { - 1 } \left( \frac { 2 } { \sqrt { } 6 } \right)\), find the values of \(a\) and \(c\).

8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\).

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2 \end{array} \right)$$ where \(\mu\) is a scalar parameter. The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)\\ (c) Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b\) where \(a\) and \(b\) are constants.\\ Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,\\ (d) determine the value of \(a\) and the value of \(b\).

1. Water is flowing into and out of a large tank.

Initially the tank contains 10 litres of water.\\ The rate of flow of the water is modelled so that

• there are \(V\) litres of water in the tank at time \(t\) minutes after the water begins to flow
• water enters the tank at a rate of \(\left( 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } \right)\) litres per minute
• water leaves the tank at a rate proportional to the volume of water remaining in the tank

Given that when \(t = 0\) the volume of water in the tank is decreasing at a rate of 3 litres per minute, use the model to\\ (a) show that the volume of water in the tank at time \(t\) satisfies $$\frac { \mathrm { d } V } { \mathrm {~d} t } = 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } - 0.4 V$$ (b) Determine \(\frac { \mathrm { d } } { \mathrm { d } t } \left( \arctan \mathrm { e } ^ { 0.4 t } \right)\) Hence, by solving the differential equation from part (a),
(c) determine an equation for the volume of water in the tank at time \(t\). Give your answer in simplest form as \(V = \mathrm { f } ( t )\) After 10 minutes, the volume of water in the tank was 8 litres.
(d) Evaluate the model in light of this information.

The plane \(\Pi_1\) has equation $$\mathbf{r} = \begin{pmatrix} 5 \\ 1 \\ 0 \end{pmatrix} + s\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix} + t\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}.$$

1. Find a cartesian equation of \(\Pi_1\). [3]

The plane \(\Pi_2\) has equation \(3x + y - z = 3\).

1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\), giving your answer in degrees. [2]
2. Find an equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\). [5]

The line \(l_1\) has equation $$\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + 3\mathbf{k})$$ and the line \(l_2\) has equation $$\mathbf{r} = 2\mathbf{i} + s\mathbf{j} + \mu(\mathbf{i} - 2\mathbf{j} + \mathbf{k})$$ where \(s\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters. Given that \(l_1\) and \(l_2\) both lie in a common plane \(\Pi_1\)

1. show that an equation for \(\Pi_1\) is \(3x + y - z = 3\) [4]
2. Find the value of \(s\). [1]

The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot (\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = 3\)

1. Find an equation for the line of intersection of \(\Pi_1\) and \(\Pi_2\) [4]
2. Find the acute angle between \(\Pi_1\) and \(\Pi_2\) giving your answer in degrees to 3 significant figures. [4]

The line \(l_1\) has equation $$\mathbf{r} = \mathbf{i} + 6\mathbf{j} - \mathbf{k} + \lambda(2\mathbf{i} + 3\mathbf{k})$$ and the line \(l_2\) has equation $$\mathbf{r} = 3\mathbf{i} + p\mathbf{j} + \mu(\mathbf{i} - 2\mathbf{j} + \mathbf{k}), \text{ where } p \text{ is a constant.}$$ The plane \(\Pi_1\) contains \(l_1\) and \(l_2\).

1. Find a vector which is normal to \(\Pi_1\). [2]
2. Show that an equation for \(\Pi_1\) is \(6x + y - 4z = 16\). [2]
3. Find the value of \(p\). [1]

The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot (\mathbf{i} + 2\mathbf{j} + \mathbf{k}) = 2\).

1. Find an equation for the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form $$(\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}.$$ [5]

A surface \(S\) has equation \(g(x, y, z) = 0\), where \(g(x, y, z) = (y - 2x)(y + z)^2 - 18\).

1. Show that \(\frac{\partial g}{\partial y} = (y + z)(-4x + 3y + z)\). [2]
2. Show that \(\frac{\partial g}{\partial x} + 2\frac{\partial g}{\partial y} - 2\frac{\partial g}{\partial z} = 0\). [4]
3. Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning. [3]
4. Find the cartesian equation of the tangent plane to the surface \(S\) at the point P\((1, 4, -7)\). [3]

The tangent plane to the surface \(S\) at the point Q\((0, 2, 1)\) has equation \(6x - 7y - 4z = -18\).

1. Find a vector equation for the line of intersection of the tangent planes at P and Q. [4]