4.04a Line equations: 2D and 3D, cartesian and vector forms

The straight line \(l\) has equation \(\mathbf{r} = 2\mathbf{i} - \mathbf{j} - 4\mathbf{k} + \lambda(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\). The plane \(p\) has equation \(3x - y + 2z = 9\). The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\). [3]
2. Find the acute angle between \(l\) and \(p\). [4]
3. Find an equation for the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(ax + by + cz = d\). [5]

The points \(A\) and \(B\) have position vectors \(\mathbf{2i - 3j + 2k}\) and \(\mathbf{5i - 2j + k}\) respectively. The plane \(p\) has equation \(x + y = 5\).

1. Find the position vector of the point of intersection of the line through \(A\) and \(B\) and the plane \(p\). [4]
2. A second plane \(q\) has an equation of the form \(x + by + cz = d\), where \(b\), \(c\) and \(d\) are constants. The plane \(q\) contains the line \(AB\), and the acute angle between the planes \(p\) and \(q\) is \(60°\). Find the equation of \(q\). [7]

Relative to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\). The line \(l\) has equation \(\mathbf{r} = 9\mathbf{i} - \mathbf{j} + 8\mathbf{k} + \mu(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\).

1. Find the position vector of the foot of the perpendicular from \(A\) to \(l\). Hence find the position vector of the reflection of \(A\) in \(l\). [5]
2. Find the equation of the plane through the origin which contains \(l\). Give your answer in the form \(ax + by + cz = d\). [3]
3. Find the exact value of the perpendicular distance of \(A\) from this plane. [3]

The equations of two lines \(l\) and \(m\) are \(\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})\) and \(\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) respectively.

1. Show that the lines do not intersect. [3]
2. Calculate the acute angle between the directions of the lines. [3]
3. Find the equation of the plane which passes through the point \((3, -2, -1)\) and which is parallel to both \(l\) and \(m\). Give your answer in the form \(ax + by + cz = d\). [5]

The line \(l\) has equation \(\mathbf{r} = 5\mathbf{i} - 3\mathbf{j} - \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + \mathbf{k})\). The plane \(p\) has equation $$(\mathbf{r} - \mathbf{i} - 2\mathbf{j}) \cdot (3\mathbf{i} + \mathbf{j} + \mathbf{k}) = 0.$$ The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\). [3]
2. Calculate the acute angle between \(l\) and \(p\). [4]
3. Find the equation of the line which lies in \(p\) and intersects \(l\) at right angles. [4]

The planes \(m\) and \(n\) have equations \(3x + y - 2z = 10\) and \(x - 2y + 2z = 5\) respectively. The line \(l\) has equation \(\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})\).

1. Show that \(l\) is parallel to \(m\). [3]
2. Calculate the acute angle between the planes \(m\) and \(n\). [3]
3. A point \(P\) lies on the line \(l\). The perpendicular distance of \(P\) from the plane \(n\) is equal to 2. Find the position vectors of the two possible positions of \(P\). [4]

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 \(l_2\).

1. Find the equation of \(\Pi_1\), giving your answer in the form \(ax + by + cz = d\). [4]

The plane \(\Pi_2\) contains \(l_2\) and the point with coordinates \((2, -1, 7)\).

1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\). [4]

The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).

1. Find a vector equation for \(PQ\). [7]

The line \(l_1\) passes through the point \(A\) with position vector \(\mathbf{i} - \mathbf{j} - 2\mathbf{k}\) and is parallel to the vector \(3\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}\). The variable line \(l_2\) passes through the point \((1 + 5 \cos t)\mathbf{i} - (1 + 5 \sin t)\mathbf{j} - 14\mathbf{k}\), where \(0 \leq t < 2\pi\), and is parallel to the vector \(15\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}\). The points \(P\) and \(Q\) are on \(l_1\) and \(l_2\) respectively, and \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).

1. Find the length of \(PQ\) in terms of \(t\). [4]
2. Hence show that the lines \(l_1\) and \(l_2\) do not intersect, and find the maximum length of \(PQ\) as \(t\) varies. [3]
3. The plane \(\Pi_1\) contains \(l_1\) and \(PQ\); the plane \(\Pi_2\) contains \(l_2\) and \(PQ\). Find the angle between the planes \(\Pi_1\) and \(\Pi_2\), correct to the nearest tenth of a degree. [4]

The position vectors of the points \(A, B, C, D\) are $$\mathbf{i} + \mathbf{j} + 3\mathbf{k}, \quad 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}, \quad -\mathbf{i} + 3\mathbf{k}, \quad m\mathbf{j} + 4\mathbf{k},$$ respectively, where \(m\) is a constant.

1. Show that the lines \(AB\) and \(CD\) are parallel when \(m = \frac{3}{2}\). [1]
2. Given that \(m \neq \frac{3}{2}\), find the shortest distance between the lines \(AB\) and \(CD\). [5]
3. When \(m = 2\), find the acute angle between the planes \(ABC\) and \(ABD\), giving your answer in degrees. [6]

With respect to a fixed origin \(O\), the line \(l_1\) is given by the equation $$\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + \lambda(8\mathbf{i} - \mathbf{j} + 4\mathbf{k})$$ where \(\lambda\) is a scalar parameter. The point \(A\) lies on \(l_1\) Given that \(|\overrightarrow{OA}| = 5\sqrt{10}\)

1. show that at \(A\) the parameter \(\lambda\) satisfies $$81\lambda^2 + 52\lambda - 220 = 0$$ [3]

Hence

1. 1. show that one possible position vector for \(A\) is \(-15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\)
   2. find the other possible position vector for \(A\). [3]

The line \(l_2\) is parallel to \(l_1\) and passes through \(O\). Given that • \(\overrightarrow{OA} = -15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\) • point \(B\) lies on \(l_2\) where \(|\overrightarrow{OB}| = 4\sqrt{10}\)

1. find the area of triangle \(OAB\), giving your answer to one decimal place. [4]

With respect to a fixed origin \(O\), the equations of lines \(l_1\) and \(l_2\) are given by $$l_1: \mathbf{r} = \begin{pmatrix} 2 \\ 8 \\ 10 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} -4 \\ -1 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 4 \\ 8 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters. Prove that lines \(l_1\) and \(l_2\) are skew. [5]

The line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$$ and the line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 0 \\ 4 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix},$$ where \(\lambda\) and \(\mu\) are parameters. The lines \(l_1\) and \(l_2\) intersect at the point \(B\) and the acute angle between \(l_1\) and \(l_2\) is \(\theta\).

1. Find the coordinates of \(B\). [4]
2. Find the value of \(\cos \theta\), giving your answer as a simplified fraction. [4]

The point \(A\), which lies on \(l_1\), has position vector \(\mathbf{a} = 3\mathbf{i} + \mathbf{j} + 2\mathbf{k}\). The point \(C\), which lies on \(l_2\), has position vector \(\mathbf{c} = 5\mathbf{i} - \mathbf{j} - 2\mathbf{k}\). The point \(D\) is such that \(ABCD\) is a parallelogram.

1. Show that \(|\overrightarrow{AB}| = |\overrightarrow{BC}|\). [3]
2. Find the position vector of the point \(D\). [2]

Referred to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \((9\mathbf{i} - 2\mathbf{j} + \mathbf{k})\), \((6\mathbf{i} + 2\mathbf{j} + 6\mathbf{k})\) and \((3\mathbf{i} + p\mathbf{j} + q\mathbf{k})\) respectively, where \(p\) and \(q\) are constants.

1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). [2]

Given that \(C\) lies on \(l\),

1. find the value of \(p\) and the value of \(q\), [2]
2. calculate, in degrees, the acute angle between \(OC\) and \(AB\). [3]

The point \(D\) lies on \(AB\) and is such that \(OD\) is perpendicular to \(AB\).

1. Find the position vector of \(D\). [6]

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 matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} k & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}, \quad k \neq 1$$

1. Show that \(\det \mathbf{M} = 2 - 2k\). [2]
2. Find \(\mathbf{M}^{-1}\), in terms of \(k\). [5] The straight line \(l_1\) is mapped onto the straight line \(l_2\) by the transformation represented by the matrix \(\begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}\). The equation of \(l_2\) is \((\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}\), where \(\mathbf{a} = 4\mathbf{i} + \mathbf{j} + 7\mathbf{k}\) and \(\mathbf{b} = 4\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).
3. Find a vector equation for the line \(l_1\). [5]

The line \(l\) passes through the point \(P(2, 1, 3)\) and is perpendicular to the plane \(\Pi\) whose vector equation is $$\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 3$$ Find

1. a vector equation of the line \(l\), [2]
2. the position vector of the point where \(l\) meets \(\Pi\). [4]
3. Hence find the perpendicular distance of \(P\) from \(\Pi\). [2]

Referred to a fixed origin \(O\), the position vectors of three non-collinear points \(A\), \(B\) and \(C\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. By considering \(\overrightarrow{AB} \times \overrightarrow{AC}\), prove that the area of \(\triangle ABC\) can be expressed in the form \(\frac{1}{2}|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|\). [5]

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]

The line \(l_1\) passes through the point \(A(0, 6, 9)\) and the point \(B(4, -6, -11)\). The line \(l_2\) has equation \(\mathbf{r} = \begin{bmatrix} -1 \\ 5 \\ -2 \end{bmatrix} + \lambda \begin{bmatrix} 3 \\ -5 \\ 1 \end{bmatrix}\).

1. The acute angle between the lines \(l_1\) and \(l_2\) is \(\theta\). Find the value of \(\cos \theta\) as a fraction in its lowest terms. [5 marks]
2. Show that the lines \(l_1\) and \(l_2\) intersect and find the coordinates of the point of intersection. [5 marks]
3. The points \(C\) and \(D\) lie on line \(l_2\) such that \(ACBD\) is a parallelogram. \includegraphics{figure_6} The length of \(AB\) is three times the length of \(CD\). Find the coordinates of the points \(C\) and \(D\). [5 marks]

Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l_1\) and \(l_2\), along which they travel are \begin{align} \mathbf{r} &= 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})
\text{and} \quad \mathbf{r} &= 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k}), \end{align} where \(\lambda\) and \(\mu\) are scalars.

1. Show that the submarines are moving in perpendicular directions. [2]
2. Given that \(l_1\) and \(l_2\) intersect at the point \(A\), find the position vector of \(A\). [5]

The point \(B\) has position vector \(10\mathbf{j} - 11\mathbf{k}\).

1. Show that only one of the submarines passes through the point \(B\). [3]
2. Given that 1 unit on each coordinate axis represents 100 m, find, in km, the distance \(AB\). [2]

Referred to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \((\mathbf{9i} - \mathbf{2j} + \mathbf{k})\), \((\mathbf{6i} + \mathbf{2j} + \mathbf{6k})\) and \((\mathbf{3i} + p\mathbf{j} + q\mathbf{k})\) respectively, where \(p\) and \(q\) are constants.

1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). [2]

Given that \(C\) lies on \(l\),

1. find the value of \(p\) and the value of \(q\), [2]
2. calculate, in degrees, the acute angle between \(OC\) and \(AB\). [3]

The point \(D\) lies on \(AB\) and is such that \(OD\) is perpendicular to \(AB\).

1. Find the position vector of \(D\). [6]

The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5\mathbf{i} + 2\mathbf{j} - 9\mathbf{k}\) and \(4\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\) respectively.

1. Find a vector equation for the line \(PQ\). [2]

The position vector of the point \(T\) is \(\mathbf{i} + 2\mathbf{j} - \mathbf{k}\).

1. Write down a vector equation for the line \(OT\) and show that \(OT\) is perpendicular to \(PQ\). [4]

It is given that \(OT\) intersects \(PQ\).

1. Find the position vector of the point of intersection of \(OT\) and \(PQ\). [3]
2. Hence find the perpendicular distance from \(O\) to \(PQ\), giving your answer in an exact form. [2]

The line \(L_1\) passes through the points \((2, -3, 1)\) and \((-1, -2, -4)\). The line \(L_2\) passes through the point \((3, 2, -9)\) and is parallel to the vector \(\mathbf{4i} - \mathbf{4j} + \mathbf{5k}\).

1. Find an equation for \(L_1\) in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [2]
2. Prove that \(L_1\) and \(L_2\) are skew. [5]

With respect to cartesian coordinates \(Oxyz\), a laser beam ABC is fired from the point A(1, 2, 4), and is reflected at point B off the plane with equation \(x + 2y - 3z = 0\), as shown in Fig. 8. A' is the point (2, 4, 1), and M is the midpoint of AA'. \includegraphics{figure_8}

1. Show that AA' is perpendicular to the plane \(x + 2y - 3z = 0\), and that M lies in the plane. [4]

The vector equation of the line AB is \(\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\).

1. Find the coordinates of B, and a vector equation of the line A'B. [6]
2. Given that A'BC is a straight line, find the angle \(\theta\). [4]
3. Find the coordinates of the point where BC crosses the \(Oxz\) plane (the plane containing the \(x\)- and \(z\)-axes). [3]