Vectors, Constructions and Loci

Calculate \(\begin{pmatrix} 4 \\ -2 \end{pmatrix} - \begin{pmatrix} 1 \\ 3 \end{pmatrix}\). [2 marks]

The vectors \(\mathbf{m}\) and \(\mathbf{n}\) are drawn on the grid. (a) Write \(\mathbf{m}\) and \(\mathbf{n}\) as column vectors. [2 marks] (b) Calculate \(\mathbf{m} + 2\mathbf{n}\). [2 marks]

\(A\) is the point \((5, -2)\) and \(B\) is the point \((-1, 4)\). (a) Calculate \(\overrightarrow{AB}\) as a column vector. [2 marks] (b) Write down \(\overrightarrow{BA}\) as a column vector. [1 mark]

\(\mathbf{u} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}\). (a) Calculate \(3\mathbf{u} - \mathbf{v}\). [2 marks] (b) Calculate \(2\mathbf{u} + \mathbf{v}\). [1 mark]

Show that the vectors \(\begin{pmatrix} -4 \\ 6 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\) are parallel. [2 marks]

(a) Calculate the length of the vector \(\begin{pmatrix} 5 \\ 12 \end{pmatrix}\). [2 marks] (b) Calculate the length of the vector \(\begin{pmatrix} 8 \\ 6 \end{pmatrix}\). [1 mark]

What does the column vector \(\begin{pmatrix} -2 \\ 5 \end{pmatrix}\) mean?

1. A2 right and 5 up
2. B2 left and 5 up
3. C5 left and 2 up
4. D2 left and 5 down

\(A\) is \((2, 5)\) and \(B\) is \((6, 2)\). What is \(\overrightarrow{AB}\)?

1. A\(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\)
2. B\(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\)
3. C\(\begin{pmatrix} 4 \\ 3 \end{pmatrix}\)
4. D\(\begin{pmatrix} 8 \\ 7 \end{pmatrix}\)

What is \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ 4 \end{pmatrix}\)?

1. A\(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\)
2. B\(\begin{pmatrix} 3 \\ 8 \end{pmatrix}\)
3. C\(\begin{pmatrix} 10 \\ 10 \end{pmatrix}\)
4. D\(\begin{pmatrix} 4 \\ 6 \end{pmatrix}\)

What is \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} - \begin{pmatrix} 1 \\ 4 \end{pmatrix}\)?

1. A\(\begin{pmatrix} 4 \\ 6 \end{pmatrix}\)
2. B\(\begin{pmatrix} -2 \\ 2 \end{pmatrix}\)
3. C\(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\)
4. D\(\begin{pmatrix} 2 \\ 2 \end{pmatrix}\)

What is \(3\begin{pmatrix} 2 \\ -1 \end{pmatrix}\)?

1. A\(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
2. B\(\begin{pmatrix} 6 \\ -3 \end{pmatrix}\)
3. C\(\begin{pmatrix} 6 \\ -1 \end{pmatrix}\)
4. D\(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\)

\(\mathbf{p} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\). What is \(2\mathbf{p} - \mathbf{q}\)?

1. A\(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
2. B\(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)
3. C\(\begin{pmatrix} 3 \\ 10 \end{pmatrix}\)
4. D\(\begin{pmatrix} 5 \\ 10 \end{pmatrix}\)

Which vector is parallel to \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\)?

1. A\(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)
2. B\(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\)
3. C\(\begin{pmatrix} 4 \\ 5 \end{pmatrix}\)
4. D\(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\)

\(\overrightarrow{AB} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\). What is \(\overrightarrow{BA}\)?

1. A\(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\)
2. B\(\begin{pmatrix} 3 \\ -4 \end{pmatrix}\)
3. C\(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\)
4. D\(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\)

What is the length of the vector \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\)?

1. A\(7\)
2. B\(5\)
3. C\(25\)
4. D\(12\)

In the triangle \(OAB\), \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). Write down (a) \(\overrightarrow{BO}\), (b) \(\overrightarrow{BA}\). [2 marks]

\(OAB\) is a triangle with \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). \(P\) is the point on \(AB\) such that \(AP : PB = 2 : 1\). (a) Write down \(\overrightarrow{AB}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). [1 mark] (b) Find \(\overrightarrow{OP}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). Give your answer in its simplest form. [3 marks]

\(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = 4\mathbf{b} - 3\mathbf{a}\). Prove that \(A\), \(B\) and \(C\) lie on a straight line. [3 marks]

\(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). \(Q\) is the point on \(OA\) such that \(OQ : QA = 1 : 3\). Find \(\overrightarrow{BQ}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). [3 marks]

\(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). \(X\) is the point such that \(\overrightarrow{OX} = 2\mathbf{a}\), and \(Y\) is the point such that \(\overrightarrow{OY} = 2\mathbf{b}\). Show that \(XY\) is parallel to \(AB\). [3 marks]

\(\overrightarrow{PQ} = \begin{pmatrix} 6 \\ 8 \end{pmatrix}\) and \(\overrightarrow{QR} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\). (a) Show that \(P\), \(Q\) and \(R\) lie on a straight line. [2 marks] (b) Calculate the length of \(PR\). [1 mark]

\(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). What is \(\overrightarrow{AB}\)?

1. A\(\mathbf{a} - \mathbf{b}\)
2. B\(\mathbf{a} + \mathbf{b}\)
3. C\(\mathbf{b} - \mathbf{a}\)
4. D\(-\mathbf{a} - \mathbf{b}\)

\(\overrightarrow{OA} = \mathbf{a}\). What is \(\overrightarrow{AO}\)?

1. A\(\mathbf{a}\)
2. B\(-\mathbf{a}\)
3. C\(2\mathbf{a}\)
4. D\(\dfrac{1}{2}\mathbf{a}\)

\(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). \(M\) is the midpoint of \(AB\). What is \(\overrightarrow{OM}\)?

1. A\(\dfrac{1}{2}(\mathbf{a} + \mathbf{b})\)
2. B\(\dfrac{1}{2}(\mathbf{b} - \mathbf{a})\)
3. C\(\mathbf{a} + \mathbf{b}\)
4. D\(\dfrac{1}{2}\mathbf{a}\)

\(P\) is on \(AB\) with \(AP : PB = 1 : 2\). What fraction of \(AB\) is \(AP\)?

1. A\(\dfrac{1}{2}\)
2. B\(\dfrac{2}{3}\)
3. C\(\dfrac{1}{4}\)
4. D\(\dfrac{1}{3}\)

\(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\), and \(P\) is on \(AB\) with \(AP : PB = 1 : 2\). What is \(\overrightarrow{OP}\)?

1. A\(\dfrac{1}{3}\mathbf{a} + \dfrac{2}{3}\mathbf{b}\)
2. B\(\dfrac{1}{3}(\mathbf{b} - \mathbf{a})\)
3. C\(\dfrac{2}{3}\mathbf{a} + \dfrac{1}{3}\mathbf{b}\)
4. D\(\mathbf{a} + \dfrac{1}{3}\mathbf{b}\)

How do you show that two lines are parallel using vectors?

1. AShow that the vectors have the same length
2. BShow that one vector is a multiple of the other
3. CShow that the vectors add to zero
4. DShow that the vectors meet at a point

\(\overrightarrow{AB} = 2\overrightarrow{BC}\). What does this show?

1. A\(A\), \(B\) and \(C\) are on a straight line
2. B\(B\) is the midpoint of \(AC\)
3. C\(AB\) is perpendicular to \(BC\)
4. D\(ABC\) is an isosceles triangle

\(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = 3\mathbf{b} - 2\mathbf{a}\). What is \(\overrightarrow{BC}\)?

1. A\(3\mathbf{b} - 2\mathbf{a}\)
2. B\(2\mathbf{a} - 2\mathbf{b}\)
3. C\(4\mathbf{b} - 2\mathbf{a}\)
4. D\(2\mathbf{b} - 2\mathbf{a}\)

What is the length of the vector \(\begin{pmatrix} 5 \\ 12 \end{pmatrix}\)?

1. A\(17\)
2. B\(7\)
3. C\(13\)
4. D\(60\)

Use ruler and compasses to construct a triangle \(ABC\) with \(AB = 7\) cm, \(BC = 6\) cm and \(AC = 5\) cm. You must show all your construction lines. [3 marks]

The diagram shows a straight line \(l\) and a point \(P\) that is not on the line. (a) Use ruler and compasses to construct the perpendicular from \(P\) to the line \(l\). You must show all your construction lines. [2 marks] (b) The point \(P\) is 3 cm above the line. Write down the shortest distance from \(P\) to \(l\). [1 mark]

Use ruler and compasses to construct the perpendicular bisector of a line \(XY\) of length 8 cm. You must show all your construction lines. [3 marks]

Explain why the compasses must be set to more than half the length of the line when constructing a perpendicular bisector. [2 marks]

Use ruler and compasses to construct an angle of \(60^\circ\) at the point \(A\) on a line. You must show all your construction lines. [3 marks]

Use ruler and compasses to construct an angle of \(30^\circ\) at the point \(A\) on a line. You must show all your construction lines. [4 marks]

What does the perpendicular bisector of a line do?

1. ACuts it in half at any angle
2. BIs parallel to it
3. CCuts it in a ratio of \(2 : 1\)
4. DCuts it in half at right angles

What must you leave on your drawing in a construction?

1. ANothing, rub them out
2. BOnly the final line
3. CThe construction arcs
4. DThe protractor marks

Which instruments do you use for a construction?

1. AA protractor only
2. BA ruler and compasses
3. CA calculator
4. DA set square and a protractor

Which angle is constructed using two arcs of the same radius, as in an equilateral triangle?

1. A\(60^\circ\)
2. B\(45^\circ\)
3. C\(90^\circ\)
4. D\(30^\circ\)

A \(60^\circ\) angle is bisected. What is the size of each part?

1. A\(15^\circ\)
2. B\(60^\circ\)
3. C\(120^\circ\)
4. D\(30^\circ\)

Why must the compasses be opened to more than half the length of the line when constructing a perpendicular bisector?

1. ASo that the line is longer
2. BSo that the angle is \(90^\circ\)
3. CSo that the arcs from both ends cross
4. DIt does not matter

What is true of every point on an angle bisector?

1. AIt is on the vertex
2. BIt is the same distance from both arms
3. CIt is perpendicular to both arms
4. DIt is the same distance from the vertex

A triangle has sides 6 cm, 5 cm and 4 cm. After drawing the 6 cm side \(AB\), how do you find \(C\) if \(AC = 4\) cm and \(BC = 5\) cm?

1. AArc of radius 4 cm from \(A\) and arc of radius 5 cm from \(B\), where they cross
2. BArc of radius 5 cm from \(A\) and arc of radius 4 cm from \(A\)
3. CA line of length 9 cm
4. DA \(90^\circ\) angle at \(A\)

What is the shortest distance from a point to a line?

1. AThe distance to the nearest end
2. BThe distance along a \(45^\circ\) line
3. CThe distance to the midpoint
4. DThe perpendicular distance

Draw the locus of all the points that are 2 cm from a point \(T\). [2 marks]

The diagram shows the line \(XY\), which is 6 cm long. Shade the region of points that are less than 2 cm from the line \(XY\) and closer to \(X\) than to \(Y\). [4 marks]

A square garden has sides of 10 m. A path must be the same distance from two adjacent sides, \(AB\) and \(AD\), starting at the corner \(A\). Describe the locus of points inside the garden that are the same distance from \(AB\) and \(AD\). [3 marks]

A radio mast transmits to a distance of 30 km. A scale drawing uses 1 cm for 10 km. Describe the locus of the points that are exactly 30 km from the mast, and say how to draw it. [3 marks]

\(A\) and \(B\) are two points 8 cm apart. Show how to find the region of points that are less than 5 cm from \(A\) and closer to \(B\) than to \(A\). [4 marks]

\(P\) and \(Q\) are 12 cm apart. Is there a point that is closer to \(Q\) than to \(P\) and less than 5 cm from \(P\)? Explain your answer. [3 marks]

What is the locus of points 3 cm from a point \(P\)?

1. AA circle with centre \(P\) and radius 3 cm
2. BA straight line 3 cm long
3. CA circle with diameter 3 cm
4. DA square with sides 3 cm

What is the locus of points that are the same distance from two points \(A\) and \(B\)?

1. AA circle through \(A\) and \(B\)
2. BThe line \(AB\)
3. CThe angle bisector at \(A\)
4. DThe perpendicular bisector of \(AB\)

What is the locus of points that are the same distance from two lines that meet at a point?

1. AA circle
2. BThe perpendicular bisector of the lines
3. CThe bisector of the angle between the lines
4. DA line parallel to both

On a drawing, which region is “less than 3 cm from \(P\)”?

1. AOutside the circle of radius 3 cm around \(P\)
2. BInside the circle of radius 3 cm around \(P\)
3. COn the circle only
4. DA straight line through \(P\)

What is the locus of points 2 cm from a line segment?

1. AParallel lines 2 cm on each side with semicircular ends
2. BA circle of radius 2 cm
3. CA single parallel line
4. DA rectangle with no rounded ends

\(A\) and \(B\) are two points. Which side of the perpendicular bisector is “closer to \(A\) than \(B\)”?

1. AThe side containing \(B\)
2. BBoth sides
3. CNeither side
4. DThe side containing \(A\)

On a scale drawing with 1 cm for 1 m, a tree must be within 4 m of a post. What do you draw?

1. AA circle of radius 4 m around the post
2. BA line 4 cm long
3. CA circle of radius 4 cm around the post
4. DA circle of radius 1 cm

A rectangle \(ABCD\) has \(A\) at the bottom left and \(B\) to its right. Which line separates the points closer to \(AB\) from those closer to \(AD\)?

1. AThe diagonal \(BD\)
2. BThe bisector of angle \(A\)
3. CThe perpendicular bisector of \(AB\)
4. DThe line \(AC\)

\(A\) and \(B\) are 8 cm apart. Is there a point closer to \(B\) than \(A\) that is less than 3 cm from \(A\)?

1. ANo
2. BYes, any point inside the circle
3. CYes, on the line \(AB\)
4. DOnly on the circle

Write down the three-figure bearing of (a) west, (b) north-east. [2 marks]

The diagram shows two points \(A\) and \(B\). On a map, \(AB\) is 4 cm and the scale is 1 : 50 000. The bearing of \(B\) from \(A\) is \(240^\circ\). (a) Calculate the bearing of \(A\) from \(B\). [2 marks] (b) Calculate the real distance \(AB\) in kilometres. [2 marks]

The bearing of \(B\) from \(A\) is \(250^\circ\). Calculate the bearing of \(A\) from \(B\). [2 marks]

A plan is drawn with a scale of 1 : 200. [3 marks] (a) A wall is 6 cm long on the plan. Calculate its real length in metres. [2 marks] (b) A room is 8 m long. Calculate its length on the plan. [1 mark]

A triangular prism has a base of 6 cm and a height of 4 cm in its triangular face, and the prism is 10 cm long. It lies on a rectangular face. Write down the dimensions of (a) its plan, (b) its elevation from the end, (c) its elevation from the side. [3 marks]

\(B\) is on a bearing of \(090^\circ\) from \(A\), and \(AB = 8\) km. \(C\) is on a bearing of \(180^\circ\) from \(B\), and \(BC = 6\) km. (a) Calculate the length \(AC\). [2 marks] (b) Write down the bearing of \(A\) from \(B\). [1 mark] (c) Write down the angle \(ABC\). [1 mark]

What is the bearing of east?

1. A\(000^\circ\)
2. B\(090^\circ\)
3. C\(180^\circ\)
4. D\(270^\circ\)

What is the bearing of south?

1. A\(180^\circ\)
2. B\(090^\circ\)
3. C\(270^\circ\)
4. D\(360^\circ\)

The bearing of \(B\) from \(A\) is \(130^\circ\). What is the bearing of \(A\) from \(B\)?

1. A\(050^\circ\)
2. B\(130^\circ\)
3. C\(230^\circ\)
4. D\(310^\circ\)

The bearing of \(B\) from \(A\) is \(072^\circ\). What is the bearing of \(A\) from \(B\)?

1. A\(108^\circ\)
2. B\(072^\circ\)
3. C\(252^\circ\)
4. D\(288^\circ\)

The bearing of \(B\) from \(A\) is \(250^\circ\). What is the bearing of \(A\) from \(B\)?

1. A\(110^\circ\)
2. B\(070^\circ\)
3. C\(430^\circ\)
4. D\(250^\circ\)

From where, and in which direction, is a bearing measured?

1. AClockwise from the north line
2. BAnticlockwise from north
3. CClockwise from east
4. DAnticlockwise from the path

A map has a scale of \(1 : 25\,000\). Two towns are 6 cm apart on the map. What is the real distance?

1. A15 km
2. B150 km
3. C150 m
4. D1.5 km

A plan has a scale of \(1 : 1000\). A length of 5 cm on the plan is how long in real life?

1. A5 m
2. B500 m
3. C50 m
4. D5 km

What is the plan of a solid?

1. AThe view from the front
2. BThe view from above
3. CThe view from the side
4. DThe view from below