The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda (2\mathbf{i} + \mathbf{j} + \mathbf{k})\) and \(\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 9\mathbf{k} + \mu (\mathbf{i} - 4\mathbf{j} + 2\mathbf{k})\) respectively. The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).
(a) Find the equation of \(\Pi_1\), giving your answer in the form \(ax + by + cz = d\).
The plane \(\Pi_2\) contains \(l_2\) and the point with coordinates \((2, -1, 7)\).
(b) Find the acute angle between \(\Pi_1\) and \(\Pi_2\).
The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).
(c) Find a vector equation for \(PQ\).
The matrix M is given by \(\mathbf{M} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{2}\sqrt{3} & \begin{pmatrix} 14 & 0 \\ 0 & 1 \end{pmatrix} \\ \frac{1}{2}\sqrt{3} & \frac{1}{2} & \end{pmatrix}\).
The points A, B, C have position vectors
\(2\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}, \quad 2\mathbf{i} + 4\mathbf{j} - \mathbf{k}, \quad -3\mathbf{i} - 3\mathbf{j} + 4\mathbf{k},\)
respectively, relative to the origin O.
(a) Find the equation of the plane ABC, giving your answer in the form \(ax + by + cz = d\).
The point D has position vector \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).
(b) Find the perpendicular distance from D to the plane ABC.
(c) Find the shortest distance between the lines AB and CD.
The matrix M is given by \(\mathbf{M} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{2}\sqrt{3} \\ \frac{1}{2}\sqrt{3} & \frac{1}{2} \end{pmatrix} \begin{pmatrix} 14 & 0 \\ 0 & 1 \end{pmatrix}\).
(a) The matrix M represents a sequence of two geometrical transformations in the x-y plane. Give full details of each transformation, and make clear the order in which they are applied. [4]
(b) Write \(\mathbf{M}^{-1}\) as the product of two matrices, neither of which is I. [2]
\((c) Find the equations of the invariant lines, through the origin, of the transformation represented by M. [5]\)
(d) The triangle ABC in the x-y plane is transformed by M onto triangle DEF. Given that the area of triangle DEF is 28 cm2, find the area of triangle ABC. [2]
The points A, B, C have position vectors
\(2\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}, \quad 2\mathbf{i} + 4\mathbf{j} - \mathbf{k}, \quad -3\mathbf{i} - 3\mathbf{j} + 4\mathbf{k},\)
respectively, relative to the origin O.
(a) Find the equation of the plane ABC, giving your answer in the form \(ax + by + cz = d\).
The point D has position vector \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).
(b) Find the perpendicular distance from D to the plane ABC.
(c) Find the shortest distance between the lines AB and CD.
The matrix \(\mathbf{M}\) is given by \(\mathbf{M} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 7 & 0 \\ 0 & 1 \end{pmatrix}\).
(a) The matrix \(\mathbf{M}\) represents a sequence of two geometrical transformations in the \(x-y\) plane. Give full details of each transformation, and make clear the order in which they are applied.
(b) Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf{M}\).
The triangle \(DEF\) in the \(x-y\) plane is transformed by \(\mathbf{M}\) onto triangle \(PQR\).
(c) Given that the area of triangle \(PQR\) is \(35 \text{ cm}^2\), find the area of triangle \(DEF\).
The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = \mathbf{i} + 4\mathbf{j} - \mathbf{k} + \lambda (\mathbf{j} - 2\mathbf{k})\) and \(\mathbf{r} = -3\mathbf{i} + 4\mathbf{j} + \mu (\mathbf{i} + 2\mathbf{j} + \mathbf{k})\) respectively.
(a) Find the shortest distance between \(l_1\) and \(l_2\).
The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).
(b) Obtain an equation of \(\Pi_1\) in the form \(px + qy + rz = s\).
(c) The point \((1, 1, 1)\) lies on the plane \(\Pi_2\).
It is given that the line of intersection of the planes \(\Pi_1\) and \(\Pi_2\) passes through the point \((0, 0, 2)\) and is parallel to the vector \(\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\).
Obtain an equation of \(\Pi_2\) in the form \(ax + by + cz = d\).
The lines \(l_1\) and \(l_2\) have equations
\(\mathbf{r} = -2\mathbf{i} - 3\mathbf{j} - 5\mathbf{k} + \lambda(-4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k})\)
and
\(\mathbf{r} = 2\mathbf{i} - 2\mathbf{j} + 3\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + \mathbf{k})\)
respectively.
(a) Find the shortest distance between \(l_1\) and \(l_2\).
The plane \(\Pi\) contains \(l_1\) and the point with position vector \(-\mathbf{i} - 3\mathbf{j} - 4\mathbf{k}\).
(b) Find an equation of \(\Pi\), giving your answer in the form \(ax + by + cz = d\).
Let k be a constant. The matrices A, B and C are given by
\(\mathbf{A} = \begin{pmatrix} 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & -2 \\ -1 & 3 \\ 0 & 0 \end{pmatrix} \quad \text{and} \quad \mathbf{C} = \begin{pmatrix} -2 & -1 & 1 \\ 1 & 1 & 3 \end{pmatrix}.\)
It is given that A is singular.
(a) Show that \(\mathbf{CAB} = \begin{pmatrix} 3 & -7 \\ -9 & 3 \end{pmatrix}\).
(b) Find the equations of the invariant lines, through the origin, of the transformation in the x–y plane represented by CAB.
(c) The matrices D, E and F represent geometrical transformations in the x–y plane.
Given that \(\mathbf{CAB} = \mathbf{D} - 9\mathbf{EF}\), find D, E and F.
The matrix \(\mathbf{M}\) is given by \(\mathbf{M} = \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\), where \(k\) is a constant and \(k \neq 0\) and \(k \neq 1\).
(a) The matrix \(\mathbf{M}\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied. [2]
The unit square in the \(x-y\) plane is transformed by \(\mathbf{M}\) onto parallelogram \(OPQR\).
(b) Find, in terms of \(k\), the area of parallelogram \(OPQR\) and the matrix which transforms \(OPQR\) onto the unit square. [3]
(c) Show that the line through the origin with gradient \(\frac{1}{k-1}\) is invariant under the transformation in the \(x-y\) plane represented by \(\mathbf{M}\). [3]
The plane \(\Pi_1\) has equation \(\mathbf{r} = \mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda (\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}) + \mu (3\mathbf{i} - \mathbf{k})\).
(a) Find an equation for \(\Pi_1\) in the form \(ax + by + cz = d\).
The line \(l\), which does not lie in \(\Pi_1\), has equation \(\mathbf{r} = -3\mathbf{i} + \mathbf{k} + t(\mathbf{i} + \mathbf{j} + \mathbf{k})\).
(b) Show that \(l\) is parallel to \(\Pi_1\).
(c) Find the distance between \(l\) and \(\Pi_1\).
(d) The plane \(\Pi_2\) has equation \(3x + 3y + 2z = 1\).
Find a vector equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\).
The lines \(l_1\) and \(l_2\) have equations
\(\mathbf{r} = -2\mathbf{i} - 3\mathbf{j} - 5\mathbf{k} + \lambda(-4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k})\) and \(\mathbf{r} = 2\mathbf{i} - 2\mathbf{j} + 3\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + \mathbf{k})\)
respectively.
(a) Find the shortest distance between \(l_1\) and \(l_2\).
The plane \(\Pi\) contains \(l_1\) and the point with position vector \(-\mathbf{i} - 3\mathbf{j} - 4\mathbf{k}\).
(b) Find an equation of \(\Pi\), giving your answer in the form \(ax + by + cz = d\).
Let k be a constant. The matrices A, B and C are given by
\(\mathbf{A} = \begin{pmatrix} 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & -2 \\ -1 & 3 \\ 0 & 0 \end{pmatrix} \text{ and } \mathbf{C} = \begin{pmatrix} -2 & -1 \\ 1 & 1 \\ 1 & 3 \end{pmatrix}.\)
It is given that A is singular.
(a) Show that \(\mathbf{CAB} = \begin{pmatrix} 3 & -7 \\ -9 & 3 \end{pmatrix}.\)
(b) Find the equations of the invariant lines, through the origin, of the transformation in the x–y plane represented by CAB.
(c) The matrices D, E and F represent geometrical transformations in the x–y plane.
Given that \(\mathbf{CAB} = \mathbf{D} - 9\mathbf{EF},\) find D, E and F.
The matrix M is given by \(\mathbf{M} = \begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix}\), where \(a, b, c\) are real constants and \(b \neq 0\).
It is given that M represents the sequence of two transformations in the x–y plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, x-axis fixed, with (0, 1) mapped to (5, 1).
The plane \(\Pi_1\) has equation \(\mathbf{r} = -4\mathbf{j} - 3\mathbf{k} + \lambda (\mathbf{i} - \mathbf{j} + \mathbf{k}) + \mu (\mathbf{i} + \mathbf{j} - \mathbf{k})\).
The matrix \(\mathbf{M}\) is given by \(\mathbf{M} = \begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix}\), where \(a, b, c\) are real constants and \(b \neq 0\).
The plane \(\Pi_1\) has equation \(\mathbf{r} = -4\mathbf{j} - 3\mathbf{k} + \lambda (\mathbf{i} - \mathbf{j} + \mathbf{k}) + \mu (\mathbf{i} + \mathbf{j} - \mathbf{k})\).
(a) Obtain an equation of \(\Pi_1\) in the form \(px + qy + rz = d\).
(b) The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot (-5\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}) = 4\).
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 \(a\mathbf{i} + a\mathbf{j} + (a-7)\mathbf{k}\) and is parallel to \((1-b)\mathbf{i} + b\mathbf{j} + b\mathbf{k}\), where \(a\) and \(b\) are positive constants.
(c) Given that the perpendicular distance from \(A\) to \(\Pi_1\) is \(\sqrt{2}\), find the value of \(a\).
(d) Given that the obtuse angle between \(l\) and \(\Pi_1\) is \(\frac{3}{4}\pi\), find the exact value of \(b\).
The points A, B, C have position vectors \(\mathbf{i} + \mathbf{j}, \ -\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}, \ -2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\), respectively, relative to the origin O.
(a) Find the equation of the plane ABC, giving your answer in the form \(ax + by + cz = d\).
(b) Find the perpendicular distance from O to the plane ABC.
(c) Find a vector equation of the common perpendicular to the lines OC and AB.
The points A, B, C have position vectors
\(4\mathbf{i} - 4\mathbf{j} + \mathbf{k}\),\( \quad -4\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\), \(\quad 4\mathbf{i} - \mathbf{j} - 2\mathbf{k}\),
respectively, relative to the origin O.
(a) Find the equation of the plane ABC, giving your answer in the form \(ax + by + cz = d\).
(b) Find the perpendicular distance from O to the plane ABC.
(c) The point D has position vector \(2\mathbf{i} + 3\mathbf{j} - 3\mathbf{k}\).
Find the coordinates of the point of intersection of the line OD with the plane ABC.
The matrix \(A\) is given by \(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{pmatrix}\).