4.04c Scalar product: calculate and use for angles

The line \(L\) passes through the points A\((1, 2, 3)\) and B\((2, 3, 1)\).

1. 1. Find the vector \(\overrightarrow{AB}\).
   2. Write down the vector equation of the line \(L\). [3]
2. The plane \(\Pi\) has equation \(x + 3y - 2z = 5\).
   1. Find the coordinates of the point of intersection of \(L\) and \(\Pi\).
   2. Find the acute angle between \(L\) and \(\Pi\). [9]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of \(P\). [2]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [2]

1. Find the value of \(k\) such that \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ k \end{pmatrix}\) are perpendicular. [2]

Two lines have equations \(l_1: \mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}\) and \(l_2: \mathbf{r} = \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\).

1. Find the point of intersection of \(l_1\) and \(l_2\). [4]
2. The vector \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) is perpendicular to the lines \(l_1\) and \(l_2\). Find the values of \(a\) and \(b\). [5]

The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the angle between \(AB\) and \(AC\). [6]
2. Find the area of triangle \(ABC\). [2]

With respect to a fixed origin \(O\), the line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ 8 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}, \text{ where } \lambda \text{ is a scalar parameter.}$$ The point \(A\) lies on \(l\) and has coordinates \((3, -2, 6)\). The point \(P\) has position vector \((-\mathbf{i} + 2\mathbf{k})\) relative to \(O\). Given that vector \(\overrightarrow{PA}\) is perpendicular to \(l\), and that point \(B\) is a point on \(l\) such that \(\angle BPA = 45°\), find the coordinates of the two possible positions of \(B\). [5]

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations \begin{align} l_1: \quad \mathbf{r} &= (-9\mathbf{i} + 10\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})
l_2: \quad \mathbf{r} &= (3\mathbf{i} + \mathbf{j} + 17\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k}) \end{align} where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l_1\) and \(l_2\) meet and find the position vector of their point of intersection. [6]
2. Show that \(l_1\) and \(l_2\) are perpendicular to each other. [2]

The point \(A\) has position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\).

1. Show that \(A\) lies on \(l_1\). [1]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -1 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ P is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of P. [3]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]

Q is a point on \(l_1\) which is 12 metres away from P. R is the point on \(l_2\) such that QR is perpendicular to \(l_1\).

1. Determine the length QR. [2]

The plane \(\Pi\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that vector \(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) is perpendicular to \(\Pi\). [2]
2. Hence find a Cartesian equation of \(\Pi\). [2]

The line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 6 \\ -3 \end{pmatrix}$$ where \(t\) is a scalar parameter. The point \(A\) lies on \(l\). Given that the shortest distance between \(A\) and \(\Pi\) is \(2\sqrt{29}\)

1. determine the possible coordinates of \(A\). [4]

The points \(A\) and \(B\) have position vectors \(5\mathbf{j} + 11\mathbf{k}\) and \(c\mathbf{i} + d\mathbf{j} + 21\mathbf{k}\) respectively, where \(c\) and \(d\) are constants. The line \(l\), through the points \(A\) and \(B\), has vector equation \(\mathbf{r} = 5\mathbf{j} + 11\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 5\mathbf{k})\), where \(\lambda\) is a parameter.

1. Find the value of \(c\) and the value of \(d\). [3]

The point \(P\) lies on the line \(l\), and \(\overrightarrow{OP}\) is perpendicular to \(l\), where \(O\) is the origin.

1. Find the position vector of \(P\). [6]
2. Find the area of triangle \(OAB\), giving your answer to 3 significant figures. [4]

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = (\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k})$$ $$l_2: \mathbf{r} = (2\mathbf{j} + 12\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k})$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l_1\) and \(l_2\) meet and find the position vector of their point of intersection. [6]
2. Show that \(l_1\) and \(l_2\) are perpendicular to each other. [2]

The point \(A\), with position vector \(5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}\), lies on \(l_1\) The point \(B\) is the image of \(A\) after reflection in the line \(l_2\)

1. Find the position vector of \(B\). [3]

The equations of two lines are \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \lambda(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) and \(\mathbf{r} = 6\mathbf{i} + 8\mathbf{j} + \mathbf{k} + \mu(\mathbf{i} + 4\mathbf{j} - 5\mathbf{k})\).

1. Show that these lines meet, and find the coordinates of the point of intersection. [5]
2. Find the acute angle between these lines. [3]

The three dimensional non-zero vector \(\mathbf{u}\) has the following properties:

• The angle \(\theta\) between \(\mathbf{u}\) and the vector \(\begin{pmatrix} 1 \\ 5 \\ 9 \end{pmatrix}\) is acute.
• The (non-reflex) angle between \(\mathbf{u}\) and the vector \(\begin{pmatrix} 9 \\ 5 \\ 1 \end{pmatrix}\) is \(2\theta\).
• \(\mathbf{u}\) is perpendicular to the vector \(\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\).

Find the angle \(\theta\). [5]

Line \(l_1\) has Cartesian equation $$x - 3 = \frac{2y + 2}{3} = 2 - z$$

1. Write the equation of line \(l_1\) in the form $$\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$$ where \(\lambda\) is a parameter and \(\mathbf{a}\) and \(\mathbf{b}\) are vectors to be found. [2 marks]

1. Line \(l_2\) passes through the points \(P(3, 2, 0)\) and \(Q(n, 5, n)\), where \(n\) is a constant.
   1. Show that the lines \(l_1\) and \(l_2\) are not perpendicular. [3 marks]
   2. Explain briefly why lines \(l_1\) and \(l_2\) cannot be parallel. [2 marks]
   3. Given that \(\theta\) is the acute angle between lines \(l_1\) and \(l_2\), show that $$\cos \theta = \frac{p}{\sqrt{34n^2 + qn + 306}}$$ where \(p\) and \(q\) are constants to be found. [3 marks]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of \(P\). [3]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]

\(Q\) is a point on \(l_1\) which is 12 metres away from \(P\). \(R\) is the point on \(l_2\) such that \(QR\) is perpendicular to \(l_1\).

1. Determine the length \(QR\). [2]

1. Find, in terms of \(x\), a vector which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [2]
2. Find the shortest possible vector of the form \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [5]

1. Vector \(\mathbf{v}\) is perpendicular to both \(\begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ p \\ p^2 \end{pmatrix}\) where \(p\) is a real number. Show that it is impossible for \(\mathbf{v}\) to be perpendicular to the vector \(\begin{pmatrix} 1 \\ 1 \\ p-1 \end{pmatrix}\). [6]

Line \(l_1\) has Cartesian equation $$l_1: \frac{-x}{2} = \frac{y-5}{2} = \frac{-z-6}{7}.$$

1. Find a vector equation for \(l_1\). [2]

Line \(l_2\) has vector equation $$l_2: \mathbf{r} = \begin{pmatrix} 2 \\ 7 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}.$$

1. Find the point of intersection of \(l_1\) and \(l_2\). [3]
2. Find the acute angle between \(l_1\) and \(l_2\). [3]

The points \(A\) and \(B\) are at \((2, 3, 5)\) and \((8, 2, 4)\) with respect to the origin \(O\).

1. Find the size of angle \(AOB\). [4]
2. Show that triangle \(AOB\) is isosceles. [3]

The planes \(\Pi_1\) and \(\Pi_2\) are both perpendicular to \(\mathbf{n}\), where \(\mathbf{n} = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}\). The points \(A(0, -9, 13)\) and \(B(8, 7, -3)\) lie in \(\Pi_1\) and \(\Pi_2\) respectively.

1. Find the equations of \(\Pi_1\) and \(\Pi_2\) in the form \(\mathbf{r} \cdot \mathbf{n} = d\) and show that \(\overrightarrow{AB}\) is parallel to \(\mathbf{n}\). [4]
2. Calculate the perpendicular distance between \(\Pi_1\) and \(\Pi_2\). [2]
3. Write down two vectors which are perpendicular to \(\mathbf{n}\) and hence find, in the form $$\mathbf{r} = \mathbf{u} + \lambda\mathbf{v} + \mu\mathbf{w},$$ an equation for the plane \(\Pi_3\) which is parallel to \(\Pi_1\) and \(\Pi_2\) and exactly half-way between them. [4]
4. The locus of all points \(P\) such that \(AP = BP = 12\sqrt{2}\) is denoted by \(L\).
   1. Give a full geometrical description of \(L\). [4]
   2. Using the result of part (iii), or otherwise, find a point on \(L\) which has integer coordinates. [4]

The square-based pyramid \(P\) has vertices \(A, B, C, D\) and \(E\). The position vectors of \(A, B, C\) and \(D\) are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) respectively where $$\mathbf{a} = \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 8 \\ -6 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}$$

1. Find the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AC}\), \(\overrightarrow{AD}\), \(\overrightarrow{BC}\), \(\overrightarrow{BD}\) and \(\overrightarrow{CD}\). [3]
2. Find
   1. the length of a side of the square base of \(P\),
   2. the cosine of the angle between one of the slanting edges of \(P\) and its base,
   3. the height of \(P\),
   4. the position vector of \(E\).
   [9] A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
3. Find the position vector of the other vertex of this octahedron. [3]

The line \(L\) has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ -3 \\ -8 \end{pmatrix} + t \begin{pmatrix} -5 \\ 3 \\ 4 \end{pmatrix}$$ The point \(P\) has position vector \(\begin{pmatrix} -7 \\ 2 \\ 7 \end{pmatrix}\). The point \(P'\) is the reflection of \(P\) in \(L\).

1. Find the position vector of \(P'\). [6]
2. Show that the point \(A\) with position vector \(\begin{pmatrix} -7 \\ 9 \\ 8 \end{pmatrix}\) lies on \(L\). [1]
3. Show that angle \(PAP' = 120°\). [3]

% Figure 3 shows kite APBP' with angle at A = 120° \includegraphics{figure_3} Figure 3 The point \(B\) lies on \(L\) and \(APBP'\) forms a kite as shown in Figure 3. The area of the kite is \(50\sqrt{3}\)

1. Find the position vector of the point \(B\). [5]
2. Show that angle \(BPA = 90°\). [2]

The circle \(C\) passes through the points \(A\), \(P\), \(P'\) and \(B\).

1. Find the position vector of the centre of \(C\). [2]

[Total 19 marks]