4.04c Scalar product: calculate and use for angles

\includegraphics{figure_9} The diagram shows three points \(A\), \(B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow{OA} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\), \(\overrightarrow{OB} = \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}\) and \(\overrightarrow{OC} = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}\). The point \(D\) lies on \(BC\), between \(B\) and \(C\), and is such that \(CD = 2DB\).

1. Find the equation of the plane \(ABC\), giving your answer in the form \(ax + by + cz = d\). [6]
2. Find the position vector of \(D\). [1]
3. Show that the length of the perpendicular from \(A\) to \(OD\) is \(\frac{1}{3}\sqrt{(65)}\). [4]

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 planes \(\Pi_1\) and \(\Pi_2\) have vector equations $$\mathbf{r} = \lambda_1(\mathbf{i} + \mathbf{j} - \mathbf{k}) + \mu_1(2\mathbf{i} - \mathbf{j} + \mathbf{k}) \quad \text{and} \quad \mathbf{r} = \lambda_2(\mathbf{i} + 2\mathbf{j} + \mathbf{k}) + \mu_2(3\mathbf{i} + \mathbf{j} - \mathbf{k})$$ respectively. The line \(l\) passes through the point with position vector \(4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\) and is parallel to both \(\Pi_1\) and \(\Pi_2\). Find a vector equation for \(l\). [6] Find also the shortest distance between \(l\) and the line of intersection of \(\Pi_1\) and \(\Pi_2\). [4]

Answer only one of the following two alternatives. EITHER The points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{i}\), \(2\mathbf{j}\) and \(4\mathbf{k}\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(ABC\). The point \(P\) on the line-segment \(ON\) is such that \(OP = \frac{3}{4}ON\). The line \(AP\) meets the plane \(OBC\) at \(Q\). Find a vector perpendicular to the plane \(ABC\) and show that the length of \(ON\) is \(\frac{1}{\sqrt{(21)}}\). [4] Find the position vector of the point \(Q\). [5] Show that the acute angle between the planes \(ABC\) and \(ABQ\) is \(\cos^{-1}\left(\frac{4}{5}\right)\). [5] OR The curve \(C\) has polar equation \(r = a(1 - \cos\theta)\) for \(0 \leqslant \theta < 2\pi\). Sketch \(C\). [2] Find the area of the region enclosed by the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), the half-line \(\theta = \frac{1}{3}\pi\) and the half-line \(\theta = \frac{2}{3}\pi\). [5] Show that $$\left(\frac{\mathrm{d}s}{\mathrm{d}\theta}\right)^2 = 4a^2\sin^2\left(\frac{1}{2}\theta\right),$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\). [7]

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]

With respect to a fixed origin, \(O\), the point \(A\) has position vector $$\overrightarrow{OA} = \begin{pmatrix} 7 \\ 2 \\ -5 \end{pmatrix}$$ Given that $$\overrightarrow{AB} = \begin{pmatrix} -2 \\ 4 \\ 3 \end{pmatrix}$$

1. find the coordinates of the point \(B\). [2]

The point \(C\) has position vector $$\overrightarrow{OC} = \begin{pmatrix} a \\ 5 \\ -1 \end{pmatrix}$$ where \(a\) is a constant. Given that \(\overrightarrow{OC}\) is perpendicular to \(\overrightarrow{BC}\)

1. find the possible values of \(a\). [4]

\includegraphics{figure_1} Figure 1 shows a sketch of triangle \(PQR\). Given that • \(\overrightarrow{PQ} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}\) • \(\overrightarrow{PR} = 8\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}\)

1. Find \(\overrightarrow{RQ}\) [2]
2. Find the size of angle \(PQR\), in degrees, to three significant figures. [3]

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]

With respect to a fixed origin \(O\), the line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} -9 \\ 8 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -4 \\ -3 \end{pmatrix}$$ where \(\mu\) is a scalar parameter. The point \(A\) is on \(l_1\) where \(\mu = 2\).

1. Write down the coordinates of \(A\). [1] The acute angle between \(OA\) and \(l_1\) is \(\theta\), where \(O\) is the origin.
2. Find the value of \(\cos \theta\). [3] The point \(B\) is such that \(\overrightarrow{OB} = 3\overrightarrow{OA}\). The line \(l_2\) passes through the point \(B\) and is parallel to the line \(l_1\).
3. Find a vector equation of \(l_2\). [2]
4. Find the length of \(OB\), giving your answer as a simplified surd. [1] The point \(X\) lies on \(l_2\). Given that the vector \(\overrightarrow{OX}\) is perpendicular to \(l_2\),
5. find the length of \(OX\), giving your answer to 3 significant figures. [3]

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = \begin{pmatrix} 5 \\ -3 \\ p \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 1 \\ -3 \end{pmatrix}, \quad l_2: \mathbf{r} = \begin{pmatrix} 8 \\ 5 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant. The lines \(l_1\) and \(l_2\) intersect at the point \(A\).

1. Find the coordinates of \(A\). [2]
2. Find the value of the constant \(p\). [3]
3. Find the acute angle between \(l_1\) and \(l_2\), giving your answer in degrees to 2 decimal places. [3]

The point \(B\) lies on \(l_2\) where \(\mu = 1\)

1. Find the shortest distance from the point \(B\) to the line \(l_1\), giving your answer to 3 significant figures. [3]

The line \(l_1\) has vector equation \(\mathbf{r} = \begin{pmatrix} 11 \\ 5 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ 2 \\ 4 \end{pmatrix}\), where \(\lambda\) is a parameter. The line \(l_2\) has vector equation \(\mathbf{r} = \begin{pmatrix} 24 \\ 4 \\ 13 \end{pmatrix} + \mu \begin{pmatrix} 7 \\ 1 \\ 5 \end{pmatrix}\), where \(\mu\) is a parameter.

1. Show that the lines \(l_1\) and \(l_2\) intersect. [4]
2. Find the coordinates of their point of intersection. [2]

Given that \(\theta\) is the acute angle between \(l_1\) and \(l_2\),

1. find the value of \(\cos \theta\). Give your answer in the form \(k\sqrt{3}\), where \(k\) is a simplified fraction. [4]

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 $$\mathbf{r} = 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})$$ and \(\mathbf{r} = 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k})\), 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]

Relative to a fixed origin \(O\), the point \(A\) has position vector \(4\mathbf{i} + 8\mathbf{j} - \mathbf{k}\), and the point \(B\) has position vector \(7\mathbf{i} + 14\mathbf{j} + 5\mathbf{k}\).

1. Find the vector \(\overrightarrow{AB}\). [1]
2. Calculate the cosine of \(\angle OAB\). [3]
3. Show that, for all values of \(\lambda\), the point \(P\) with position vector $$\lambda\mathbf{i} + 2\lambda\mathbf{j} + (2\lambda - 9)\mathbf{k}$$ lies on the line through \(A\) and \(B\). [3]
4. Find the value of \(\lambda\) for which \(OP\) is perpendicular to \(AB\). [3]
5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(AB\). [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 plane \(P\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix}$$

1. Find a vector perpendicular to the plane \(P\). [2] The line \(l\) passes through the point \(A(1, 3, 3)\) and meets \(P\) at \((3, 1, 2)\). The acute angle between the plane \(P\) and the line \(l\) is \(\alpha\).
2. Find \(\alpha\) to the nearest degree. [4]
3. Find the perpendicular distance from \(A\) to the plane \(P\). [4]

The plane \(\Pi\) passes through the points $$A(-1, -1, 1), B(4, 2, 1) \text{ and } C(2, 1, 0).$$

1. Find a vector equation of the line perpendicular to \(\Pi\) which passes through the point \(D(1, 2, 3)\). [3]
2. Find the volume of the tetrahedron \(ABCD\). [3]
3. Obtain the equation of \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [3]

The perpendicular from \(D\) to the plane \(\Pi\) meets \(\Pi\) at the point \(E\).

1. Find the coordinates of \(E\). [4]
2. Show that \(DE = \frac{11\sqrt{35}}{35}\). [2]

The point \(D'\) is the reflection of \(D\) in \(\Pi\).

1. Find the coordinates of \(D'\). [3]

The plane \(\Pi_1\) passes through the \(P\), with position vector \(\mathbf{i} + 2\mathbf{j} - \mathbf{k}\), and is perpendicular to the line \(L\) with equation $$\mathbf{r} = 3\mathbf{i} - 2\mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}).$$

1. Show that the Cartesian equation of \(\Pi_1\) is \(x - 5y - 3z = -6\). [4]

The plane \(\Pi_2\) contains the line \(L\) and passes through the point \(Q\), with position vector \(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\).

1. Find the perpendicular distance of \(Q\) from \(\Pi_1\). [4]
2. Find the equation of \(\Pi_2\) in the form \(\mathbf{r} = \mathbf{a} + s\mathbf{b} + t\mathbf{c}\). [4]

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]

Relative to a fixed origin \(O\), the point \(A\) has position vector \(3\mathbf{i} + 2\mathbf{j} - \mathbf{k}\), the point \(B\) has position vector \(5\mathbf{i} + \mathbf{j} + \mathbf{k}\), and the point \(C\) has position vector \(7\mathbf{i} - \mathbf{j}\).

1. Find the cosine of angle \(ABC\). [4]
2. Find the exact value of the area of triangle \(ABC\). [3]

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

1. Show that \(AC\) is perpendicular to \(CD\). [2]
2. Find the ratio \(AD:DB\). [2]

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 points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) relative to an origin \(O\), where \(\mathbf{a} = 4\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\) and \(\mathbf{b} = -7\mathbf{i} + 5\mathbf{j} + 4\mathbf{k}\).

1. Find the length of \(AB\). [3]
2. Use a scalar product to find angle \(OAB\). [3]

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]

\(ABCD\) is a parallelogram. The position vectors of \(A\), \(B\) and \(C\) are given respectively by $$\mathbf{a} = 2\mathbf{i} + \mathbf{j} + 3\mathbf{k}, \quad \mathbf{b} = 3\mathbf{i} - 2\mathbf{j}, \quad \mathbf{c} = \mathbf{i} - \mathbf{j} - 2\mathbf{k}.$$

1. Find the position vector of \(D\). [3]
2. Determine, to the nearest degree, the angle \(ABC\). [4]