The position vectors of the points A, B, C, D are
\(7\mathbf{i} + 4\mathbf{j} - \mathbf{k}, \quad 11\mathbf{i} + 3\mathbf{j}, \quad 2\mathbf{i} + 6\mathbf{j} + 3\mathbf{k}, \quad 2\mathbf{i} + 7\mathbf{j} + \lambda \mathbf{k}\)
respectively.
(a) Given that the shortest distance between the line AB and the line CD is 3, show that \(\lambda^2 - 5\lambda + 4 = 0\).
Let \(\Pi_1\) be the plane ABD when \(\lambda = 1\).
Let \(\Pi_2\) be the plane ABD when \(\lambda = 4\).
(b) (i) Write down an equation of \(\Pi_1\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + s\mathbf{b} + t\mathbf{c}\).
(ii) Find an equation of \(\Pi_2\), giving your answer in the form \(ax + by + cz = d\).
(c) Find the acute angle between \(\Pi_1\) and \(\Pi_2\).
The plane \(\Pi\) contains the lines \(\mathbf{r} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + \mathbf{k})\) and \(\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} + 2\mathbf{k} + \mu(3\mathbf{i} + 2\mathbf{j} - \mathbf{k})\).
(a) Find a Cartesian equation of \(\Pi\), giving your answer in the form \(ax + by + cz = d\). [4]
The line \(l\) passes through the point \(P\) with position vector \(2\mathbf{i} + 3\mathbf{j} + \mathbf{k}\) and is parallel to the vector \(\mathbf{j} + \mathbf{k}\).
(b) Find the acute angle between \(l\) and \(\Pi\). [3]
(c) Find the position vector of the foot of the perpendicular from \(P\) to \(\Pi\). [4]
The matrix M is given by \(M = \begin{pmatrix} \frac{1}{2}\sqrt{2} & -\frac{1}{2}\sqrt{2} \\ \frac{1}{2} & \frac{1}{2}\sqrt{2} \end{pmatrix} \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}\), where \(k\) is a constant.
(a) The matrix M represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
(b) The triangle ABC in the \(x-y\) plane is transformed by M onto triangle DEF. Find, in terms of \(k\), the single matrix which transforms triangle DEF onto triangle ABC.
(c) Find the set of values of \(k\) for which the transformation represented by M has no invariant lines through the origin.
The matrix M is given by M = \(\begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix}\), where \(k\) is a constant and \(k \neq 0\) or 1.
(a) The matrix 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]
(b) Write M-1 as the product of two matrices, neither of which is I. [2]
(c) Show that the invariant points of the transformation represented by M lie on the line \(y = \frac{k^2}{1-k}x\). [4]
(d) The triangle ABC in the x-y plane is transformed by M onto triangle DEF. Find the value of \(k\) for which the area of triangle DEF is equal to the area of triangle ABC. [2]
The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = 2\mathbf{i} + \mathbf{k} + \lambda(\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) and \(\mathbf{r} = 2\mathbf{j} + 6\mathbf{k} + \mu(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})\) respectively.
The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).
(a) Find the length \(PQ\). [5]
The plane \(\Pi_1\) contains \(PQ\) and \(l_1\).
The plane \(\Pi_2\) contains \(PQ\) and \(l_2\).
(b) (i) Write down an equation of \(\Pi_1\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + s\mathbf{b} + t\mathbf{c}\). [1]
(ii) Find an equation of \(\Pi_2\), giving your answer in the form \(ax + by + cz = d\). [4]
(c) Find the acute angle between \(\Pi_1\) and \(\Pi_2\). [5]
The matrix \(\mathbf{M}\) represents the sequence of two transformations in the \(x\)-\(y\) plane given by a rotation of \(60^\circ\) anticlockwise about the origin followed by a one-way stretch in the \(x\)-direction with scale factor \(d\) \((d \neq 0)\).
Let \(t\) be a positive constant.
The line \(l_1\) passes through the point with position vector \(t\mathbf{i} + \mathbf{j}\) and is parallel to the vector \(-2\mathbf{i} - \mathbf{j}\).
The line \(l_2\) passes through the point with position vector \(\mathbf{j} + t\mathbf{k}\) and is parallel to the vector \(-2\mathbf{j} + \mathbf{k}\).
It is given that the shortest distance between the lines \(l_1\) and \(l_2\) is \(\sqrt{21}\).
(a) Find the value of \(t\). [5]
The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).
(b) Write down an equation of \(\Pi_1\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}\).
The plane \(\Pi_2\) has Cartesian equation \(5x - 6y + 7z = 0\).
(c) Find the acute angle between \(l_2\) and \(\Pi_2\). [3]
(d) Find the acute angle between \(\Pi_1\) and \(\Pi_2\). [3]
The matrices A, B and C are given by
\(A = \begin{pmatrix} 2 & k & k \\ 5 & -1 & 3 \\ 1 & 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{pmatrix} \quad \text{and} \quad C = \begin{pmatrix} 0 & 1 & 1 \\ -1 & 2 & 0 \end{pmatrix},\)
where \(k\) is a real constant.
The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = -\mathbf{i} - 2\mathbf{j} + \mathbf{k} + s(2\mathbf{i} - 3\mathbf{j})\) and \(\mathbf{r} = 3\mathbf{i} - 2\mathbf{k} + t(3\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) respectively.
The plane \(\Pi_1\) contains \(l_1\) and the point \(P\) with position vector \(-2\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}\).
The plane \(\Pi\) has equation \(\mathbf{r} = -2\mathbf{i} + 3\mathbf{j} + 3\mathbf{k} + \lambda (\mathbf{i} + \mathbf{k}) + \mu (2\mathbf{i} + 3\mathbf{j})\).
(a) Give full details of the geometrical transformation in the x-y plane represented by the matrix \(\begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix}\).
Let \(\mathbf{A} = \begin{pmatrix} 3 & 4 \\ 2 & 2 \end{pmatrix}\).
(b) The triangle DEF in the x-y plane is transformed by \(\mathbf{A}\) onto triangle PQR. Given that the area of triangle DEF is 13 cm2, find the area of triangle PQR.
(c) Find the matrix \(\mathbf{B}\) such that \(\mathbf{AB} = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix}\).
(d) Show that the origin is the only invariant point of the transformation in the x-y plane represented by \(\mathbf{A}\).
The points A, B, C have position vectors
\(2\mathbf{i} + 2\mathbf{j}, \quad -\mathbf{j} + \mathbf{k} \quad \text{and} \quad 2\mathbf{i} + \mathbf{j} - 7\mathbf{k}\)
respectively, relative to the origin O.
(a) Find an equation of the plane OAB, giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\).
The plane \(\Pi\) has equation \(x - 3y - 2z = 1\).
(b) Find the perpendicular distance of \(\Pi\) from the origin.
(c) Find the acute angle between the planes OAB and \(\Pi\).
(d) Find an equation for the common perpendicular to the lines OC and AB.
4 The points \(A, B, C\) have position vectors
\(-\mathbf{i}+\mathbf{j}+2 \mathbf{k}, \quad-2 \mathbf{i}-\mathbf{j}, \quad 2 \mathbf{i}+2 \mathbf{k},\)
respectively, relative to the origin \(O\).
(a) Find the equation of the plane \(A B C\), giving your answer in the form \(a x+b y+c z=d\).
(b) Find the perpendicular distance from \(O\) to the plane \(A B C\).
(c) Find the acute angle between the planes \(O A B\) and \(A B C\).
5 The lines \(l_{1}\) and \(l_{2}\) have equations \(\mathbf{r}=3 \mathbf{i}+3 \mathbf{k}+\lambda(\mathbf{i}+4 \mathbf{j}+4 \mathbf{k})\) and \(\mathbf{r}=3 \mathbf{i}-5 \mathbf{j}-6 \mathbf{k}+\mu(5 \mathbf{j}+6 \mathbf{k})\) respectively.
(a) Find the shortest distance between \(l_{1}\) and \(l_{2}\).
The plane \(\Pi\) contains \(l_{1}\) and is parallel to the vector \(\mathbf{i}+\mathbf{k}\).
(b) Find the equation of \(\Pi\), giving your answer in the form \(a x+b y+c z=d\).
(c) Find the acute angle between \(l_{2}\) and \(\Pi\).
7 The line \(l_{1}\) passes through the points \(A(-3,1,4)\) and \(B(-1,5,9)\). The line \(l_{2}\) passes through the points \(C(-2,6,5)\) and \(D(-1,7,5)\).
(i) Find the shortest distance between the lines \(l_{1}\) and \(l_{2}\).
(ii) Find the acute angle between the line \(l_{2}\) and the plane containing \(A, B\) and \(D\).
With \(O\) as the origin, the points \(A, B, C\) have position vectors
\(\mathbf{i}-\mathbf{j}, \quad 2 \mathbf{i}+\mathbf{j}+7 \mathbf{k}, \quad \mathbf{i}-\mathbf{j}+\mathbf{k}\)
respectively.
(i) Find the shortest distance between the lines \(O C\) and \(A B\).
(ii) Find the cartesian equation of the plane containing the line \(O C\) and the common perpendicular of the lines \(O C\) and \(A B\).
The line \(l_1\) is parallel to the vector \(a\mathbf i-\mathbf j+\mathbf k\), where \(a\) is a constant, and passes through the point whose position vector is \(9\mathbf j+2\mathbf k\). The line \(l_2\) is parallel to the vector \(-a\mathbf i+2\mathbf j+4\mathbf k\) and passes through the point whose position vector is \(-6\mathbf i-5\mathbf j+10\mathbf k\).
(i) It is given that \(l_1\) and \(l_2\) intersect.
(a) Show that \(a=-\frac{6}{13}\).
(b) Find a Cartesian equation of the plane containing \(l_1\) and \(l_2\).
(ii) Given instead that the perpendicular distance between \(l_1\) and \(l_2\) is \(3\sqrt{30}\), find the value of \(a\).
The lines \(l_{1}\) and \(l_{2}\) have vector equations
\(\mathbf{r}=a \mathbf{i}+9 \mathbf{j}+13 \mathbf{k}+\lambda(\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}) \quad \text { and } \quad \mathbf{r}=-3 \mathbf{i}+7 \mathbf{j}-2 \mathbf{k}+\mu(-\mathbf{i}+2 \mathbf{j}-3 \mathbf{k})\)
respectively. It is given that \(l_{1}\) and \(l_{2}\) intersect.
(i) Find the value of the constant \(a\).
The point \(P\) has position vector \(3 \mathbf{i}+\mathbf{j}+6 \mathbf{k}\).
(ii) Find the perpendicular distance from \(P\) to the plane containing \(l_{1}\) and \(l_{2}\).
(iii) Find the perpendicular distance from \(P\) to \(l_{2}\).
The plane \(\Pi_1\) has equation
\(\mathbf r=\begin{pmatrix}5\\1\\0\end{pmatrix}+s\begin{pmatrix}-4\\1\\3\end{pmatrix}+t\begin{pmatrix}0\\1\\2\end{pmatrix}.\)
(i) Find a cartesian equation of \(\Pi_1\).
The plane \(\Pi_2\) has equation \(3x+y-z=3\).
(ii) Find the acute angle between \(\Pi_1\) and \(\Pi_2\), giving your answer in degrees.
(iii) Find an equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form \(\mathbf r=\mathbf a+\lambda\mathbf b\).
(a) The plane \(\Pi_1\) has equation \(\mathbf r=-3\mathbf i-\mathbf j-\mathbf k+\lambda(\mathbf j+2\mathbf k)+\mu(\mathbf i+3\mathbf j+\mathbf k)\). Find an equation for \(\Pi_1\) in the form \(ax+by+cz=d\).
(b) Find the perpendicular distance from the point with position vector \(-\mathbf i-2\mathbf k\) to \(\Pi_1\).
(c) The plane \(\Pi_2\) has equation \(3x+2y-z=14\). Find a vector equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\).