1.10g Problem solving with vectors: in geometry

The circle \((x - 3)^2 + (y - 2)^2 = 20\) has centre C.

1. Write down the radius of the circle and the coordinates of C. [2]
2. Find the coordinates of the intersections of the circle with the \(x\)- and \(y\)-axes. [5]
3. Show that the points A\((1, 6)\) and B\((7, 4)\) lie on the circle. Find the coordinates of the midpoint of AB. Find also the distance of the chord AB from the centre of the circle. [5]

The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).

1. Find the coordinates of \(A\). [2]

The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).

1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]

The line \(l_2\) crosses the \(x\)-axis at the point \(C\).

1. Find the coordinates of \(C\). [1]

The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).

1. Show that \(D\) has coordinates \((5, 4)\). [5]
2. Find the area of triangle \(ACD\). [2]

The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.

1. Show that \(\angle PQR = 90°\). [4]

Given that \(P\), \(Q\) and \(R\) all lie on circle \(C\),

1. find the coordinates of the centre of \(C\), [3]
2. show that the equation of \(C\) can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]

The point \(A\) has coordinates \((4, -3, 2)\). The line \(l_1\) passes through \(A\) and has equation \(\mathbf{r} = \begin{bmatrix} 4 \\ -3 \\ 2 \end{bmatrix} + \lambda \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}\). The line \(l_2\) has equation \(\mathbf{r} = \begin{bmatrix} -1 \\ 3 \\ 4 \end{bmatrix} + \mu \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix}\). The point \(B\) lies on \(l_2\) where \(\mu = 2\).

1. Find the vector \(\overrightarrow{AB}\). [3 marks]
2. 1. Show that the lines \(l_1\) and \(l_2\) intersect. [4 marks]
   2. The lines \(l_1\) and \(l_2\) intersect at the point \(P\). Find the coordinates of \(P\). [1 mark]
3. The point \(C\) lies on a line which is parallel to \(l_1\) and which passes through the point \(B\). The points \(A\), \(B\), \(C\) and \(P\) are the vertices of a parallelogram. Find the coordinates of the two possible positions of the point \(C\). [4 marks]

The points A, B and C have coordinates \(A(3, 2, -1)\), \(B(-1, 1, 2)\) and \(C(10, 5, -5)\), relative to the origin O. Show that \(\overrightarrow{OC}\) can be written in the form \(\lambda\overrightarrow{OA} + \mu\overrightarrow{OB}\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points O, A, B and C from the fact that \(\overrightarrow{OC}\) can be expressed as a combination of \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\)? [6]

Fig. 7 shows a tetrahedron ABCD. The coordinates of the vertices, with respect to axes Oxyz, are A(-3, 0, 0), B(2, 0, -2), C(0, 4, 0) and D(0, 4, 5). \includegraphics{figure_7}

1. Find the lengths of the edges AB and AC, and the size of the angle CAB. Hence calculate the area of triangle ABC. [7]
2. 1. Verify that 4i - 3j + 10k is normal to the plane ABC. [2]
   2. Hence find the equation of this plane. [2]
3. Write down a vector equation for the line through D perpendicular to the plane ABC. Hence find the point of intersection of this line with the plane ABC. [5]

The volume of a tetrahedron is \(\frac{1}{3} \times \text{area of base} \times \text{height}\).

1. Find the volume of the tetrahedron ABCD. [2]

The line \(l_1\) passes through the points \(A\) and \(B\) with position vectors \((\mathbf{3i} + \mathbf{6j} - \mathbf{8k})\) and \((\mathbf{8j} - \mathbf{6k})\) respectively, relative to a fixed origin.

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

The line \(l_2\) has vector equation $$\mathbf{r} = (-\mathbf{2i} + \mathbf{10j} + \mathbf{6k}) + \mu(\mathbf{7i} - \mathbf{4j} + \mathbf{6k}),$$ where \(\mu\) is a scalar parameter.

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

The point \(C\) lies on \(l_2\) and is such that \(AC\) is perpendicular to \(AB\).

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

\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane. A body of mass 1 kg moves under the action of a constant force \((4\mathbf{i} + 5\mathbf{j})\) N. The body moves from the point \(P\) with position vector \((-3\mathbf{i} - 15\mathbf{j})\) m to the point \(Q\) with position vector \(9\mathbf{i}\) m.

1. Find the work done by the force in moving the body from \(P\) to \(Q\). [5 marks]
2. Given that the body started from rest at \(P\), find its speed when it is at \(Q\). [5 marks]

Two forces \(\mathbf{F}_1 = (2i + j)\) N and \(\mathbf{F}_2 = (-2j - k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \(\mathbf{r}_1 = (3i + j + k)\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \(\mathbf{r}_2 = (i - 2j)\) m. A third force \(\mathbf{F}_3\) acts on the body such that \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are in equilibrium.

1. Find the magnitude of \(\mathbf{F}_3\). [4]

1. Find a vector equation of the line of action of \(\mathbf{F}_3\). [8]

The force \(\mathbf{F}_3\) is replaced by a fourth force \(\mathbf{F}_4\), acting through the origin \(O\), such that \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_4\) are equivalent to a couple.

1. Find the magnitude of this couple. [4]

Two forces \(\mathbf{F}_1 = (i + 2j + 3k)\) N and \(\mathbf{F}_2 = (3i + j + 2k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts through the point with position vector \((2i + k)\) m and the force \(\mathbf{F}_2\) acts through the point with position vector \((j + 2k)\) m.

1. If the two forces are equivalent to a single force \(\mathbf{R}\), find
   1. \(\mathbf{R}\), [2]
   2. a vector equation of the line of action of \(\mathbf{R}\), in the form \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}\). [6]

1. If the two forces are equivalent to a single force acting through the point with position vector \((i + 2j + k)\) m together with a couple of moment \(\mathbf{G}\), find the magnitude of \(\mathbf{G}\). [5]

Two forces \(\mathbf{F}_1 = (3i + k)\) N and \(\mathbf{F}_2 = (4i + j - k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \((2i - j + 3k)\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \((-3i + 2k)\) m. The two forces are equivalent to a single force \(\mathbf{R}\) acting at the point with position vector \((i + 2j + k)\) m together with a couple of moment \(\mathbf{G}\). Find,

1. \(\mathbf{R}\), [2]
2. \(\mathbf{G}\). [4]

A third force \(\mathbf{F}_3\) is now added to the system. The force \(\mathbf{F}_3\) acts at the point with position vector \((2i - k)\) m and the three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are equivalent to a couple.

1. Find the magnitude of the couple. [6]

A small bead is threaded on a smooth, straight horizontal wire which passes through the point \(A(-3, 1)\) and the point \(B(2, 5)\) in the \(x\)-\(y\) plane. The bead moves under the action of a horizontal force \(\mathbf{F}\) of magnitude \(8.5\) N whose line of action is parallel to the line with equation \(15x - 8y + 4 = 0\). The unit on both the \(x\) and \(y\) axes has length one metre. Find the work done by \(\mathbf{F}\) as it moves the bead from \(A\) to \(B\). [8]

Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) act on a rigid body at the points with position vectors \(\mathbf{r}_1\), \(\mathbf{r}_2\) and \(\mathbf{r}_3\) respectively. \(\mathbf{F}_1 = (2\mathbf{i} + 3\mathbf{j} - \mathbf{k})\) N and \(\mathbf{r}_1 = (\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) m, \(\mathbf{F}_2 = (\mathbf{i} - 4\mathbf{j} - 2\mathbf{k})\) N and \(\mathbf{r}_2 = (3\mathbf{i} - \mathbf{j} - \mathbf{k})\) m, \(\mathbf{F}_3 = (-3\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) N and \(\mathbf{r}_3 = (\mathbf{i} - 2\mathbf{j} + \mathbf{k})\) m. Show that the system is equivalent to a couple and find the magnitude of the vector moment of this couple. [9]

A bead of mass 0.125 kg is threaded on a smooth straight horizontal wire. The bead moves from rest at the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j} - \mathbf{k})\) m relative to a fixed origin \(O\) to a point with position vector \((3\mathbf{i} - 4\mathbf{j} - \mathbf{k})\) m relative to \(O\) under the action of a force \(\mathbf{F} = (14\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) N. Find

1. the work done by \(\mathbf{F}\) as the bead moves from \(A\) to \(B\), [3]
2. the speed of the bead at \(B\). [2]

Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) and a couple \(\mathbf{G}\) act on a rigid body. The force \(\mathbf{F}_1 = (3\mathbf{i} + 4\mathbf{j})\) N acts through the point with position vector \(2\mathbf{i}\) m relative to a fixed origin \(O\). The force \(\mathbf{F}_2 = (2\mathbf{i} - \mathbf{j} + \mathbf{k})\) N acts through the point with position vector \((\mathbf{i} + \mathbf{j})\) m relative to \(O\). The forces and couple are equivalent to a single force \(\mathbf{F}\) acting through \(O\).

1. Find \(\mathbf{F}\). [2]
2. Find \(\mathbf{G}\). [5]

(In this question, the notation \(\Delta ABC\) denotes the area of the triangle \(ABC\).) The points \(P\), \(Q\) and \(R\) have position vectors \(p\mathbf{i}\), \(q\mathbf{j}\) and \(r\mathbf{k}\) respectively, relative to the origin \(O\), where \(p\), \(q\) and \(r\) are positive. The points \(O\), \(P\), \(Q\) and \(R\) are joined to form a tetrahedron.

1. Draw a sketch of the tetrahedron and write down the values of \(\Delta OPQ\), \(\Delta OQR\) and \(\Delta ORP\). [3]
2. Use the definition of the vector product to show that \(\frac{1}{2}|\overrightarrow{RP} \times \overrightarrow{RQ}| = \Delta PQR\). [1]
3. Show that \((\Delta OPQ)^2 + (\Delta OQR)^2 + (\Delta ORP)^2 = (\Delta PQR)^2\). [6]

Points \(A(-7, -7)\), \(B(8, -1)\), \(C(4, 9)\) and \(D(-11, 3)\) are the vertices of a quadrilateral \(ABCD\).

1. Prove that \(ABCD\) is a rectangle. [4 marks]
2. Find the area of \(ABCD\). [2 marks]

A quadrilateral has vertices A, B, C and D with position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix}, \overrightarrow{OB} = \begin{pmatrix} -1 \\ 2 \\ 7 \end{pmatrix}, \overrightarrow{OC} = \begin{pmatrix} 0 \\ 7 \\ 6 \end{pmatrix} \text{ and } \overrightarrow{OD} = \begin{pmatrix} 4 \\ 10 \\ 0 \end{pmatrix}$$

1. Write down the vector \(\overrightarrow{AB}\) [1 mark]
2. Show that ABCD is a parallelogram, but not a rhombus. [5 marks]

Two particles \(P\) and \(Q\) are moving in separate straight lines across a smooth horizontal surface. \(P\) moves with constant velocity \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) \(Q\) moves from position vector \((5\mathbf{i} - 7\mathbf{j})\) metres to position vector \((14\mathbf{i} + 5\mathbf{j})\) metres during a 3 second period.

1. Show that \(P\) and \(Q\) move along parallel lines. [3 marks]
2. Stevie says Q is also moving with a constant velocity of \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) Explain why Stevie may be incorrect. [1 mark]
3. A third particle \(R\) is moving with a constant speed of 4 m s\(^{-1}\), in a straight line, across the same surface. \(P\) and \(R\) move along lines that intersect at a fixed point \(X\) It is given that: • \(P\) passes through \(X\) exactly 2 seconds after \(R\) passes through \(X\) • \(P\) and \(R\) are exactly 13 metres apart 3 seconds after \(R\) passes through \(X\) Show that \(P\) and \(R\) move along perpendicular lines. [5 marks]

\(OABC\) is a parallelogram with \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OC} = \mathbf{c}\). \(P\) is the midpoint of \(AC\). \includegraphics{figure_7}

1. Find the following in terms of \(\mathbf{a}\) and \(\mathbf{c}\), simplifying your answers.
   1. \(\overrightarrow{AC}\) [1]
   2. \(\overrightarrow{OP}\) [2]
2. Hence prove that the diagonals of a parallelogram bisect one another. [4]

Vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are given by $$\overrightarrow{AB} = \begin{pmatrix} 2p \\ q \\ 4 \end{pmatrix} \quad \overrightarrow{BC} = \begin{pmatrix} q \\ -3p \\ 2 \end{pmatrix},$$ where \(p\) and \(q\) are constants. Given that \(\overrightarrow{AC}\) is parallel to \(\begin{pmatrix} 3 \\ -4 \\ 3 \end{pmatrix}\), find the value of \(p\) and the value of \(q\). [3]

Relative to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(4\mathbf{i} + 2\mathbf{j}\), \(3\mathbf{i} + 4\mathbf{j}\) and \(-\mathbf{i} + 12\mathbf{j}\), respectively.

1. Find the magnitude of the vector \(\overrightarrow{OC}\) [2]
2. Find the angle that the vector \(\overrightarrow{OB}\) makes with the vector \(\mathbf{j}\) to the nearest degree [2]
3. Show that the points \(A\), \(B\) and \(C\) are collinear [3]

Points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\). Point \(C\) lies on \(AB\) such that \(AC : CB = p : 1\).

1. Show that the position vector of \(C\) is \(\frac{1}{p+1}(\mathbf{a} + p\mathbf{b})\). [3]

It is now given that \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) and \(\mathbf{b} = -6\mathbf{i} + 4\mathbf{j} + 12\mathbf{k}\), and that \(C\) lies on the \(y\)-axis.

1. Find the value of \(p\). [4]
2. Write down the position vector of \(C\). [1]

The points \(A\) and \(B\) have position vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\) respectively, relative to the origin \(O\). The point \(C\) is on the line \(OA\) extended so that \(\overrightarrow{AC} = 2\overrightarrow{OA}\) and the point \(D\) is on the line \(OB\) extended so that \(\overrightarrow{BD} = 3\overrightarrow{OB}\). The point \(X\) is such that \(OCXD\) is a parallelogram.

1. Show that a vector equation of the line \(AX\) is \(\mathbf{r} = \mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(5\mathbf{i} + 7\mathbf{k})\) and find an equation of the line \(CD\) in a similar form. [5]
2. Prove that the lines \(AX\) and \(CD\) intersect and find the position vector of their point of intersection. [4]

The points \(A\), \(B\) and \(C\) lie in a vertical plane and have position vectors \(4\mathbf{i}\), \(3\mathbf{j}\) and \(7\mathbf{i} + 4\mathbf{j}\), respectively. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertically upwards, respectively. The units of the components are metres.

1. Show that angle \(BAC\) is a right angle. [2]

\includegraphics{figure_10} Strings \(AB\) and \(AC\) are attached to \(B\) and \(C\), and joined at \(A\). A particle of weight 20 N is attached at \(A\) (see diagram). The particle is in equilibrium.

1. By resolving in the directions \(AB\) and \(AC\), determine the magnitude of the tension in each string. [3]
2. Express the tension in the string \(AB\) as a vector, in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [3]