Course

Cambridge O Level Mathematics: Transformations and Vectors (4024)

Name: Cambridge O Level Mathematics: Transformations and Vectors (4024)
Price: 999 BDT

Section 7 in the syllabus order: reflection, rotation, enlargement and translation, then column vectors, magnitude and vector geometry proofs.

48 lessons8h 36mEnglish

What you'll learn

Draw the image of a shape after a reflection, rotation, enlargement or translation
Describe a single transformation fully, with its mirror line, centre, angle, scale factor or vector
Carry out two transformations in order and name the single transformation that does the same
Add, subtract and scale column vectors, and write them as AB or a
Calculate the magnitude of a vector, exactly or to 3 significant figures
Use position vectors to show that lines are parallel, that points are collinear, and to solve ratio problems

Requirements

Coordinates and plotting points in all four quadrants
Straight-line equations such as x = 3, y = -1 and y = x, as taught in our Graphs course
Pythagoras' theorem and square roots, as taught in our Trigonometry course
Simplifying ratios and working with fractions

Course content

48 lessons · 8h 36m

7.1 · Transformations

Reflection, rotation, enlargement and translation on a grid: draw each one, describe it fully, and combine two of them.

0/16

Reflection in a vertical or horizontal line

Each image point is the same distance from the mirror line as the object point, on the other side. Start with the axes, x = a and y = b.

Preview12m 2.

Reflection in the lines y = x and y = -x

In y = x the two coordinates swap places. In y = -x they swap and both change sign.

Preview12m

Describing a reflection: finding the mirror line

The mirror line passes through the midpoint of each object point and its image. Name it with its equation.

12m

Rotation through multiples of 90° about the origin

A rotation turns every point about one fixed centre. Use tracing paper for 90° clockwise, 90° anticlockwise and 180°.

12m

Rotation about a centre

The centre need not be the origin. Count the steps from the centre to each point, then turn those steps.

12m

Describing a rotation: centre, angle and direction

A rotation is described by its centre, its angle and its direction. Find the centre where the perpendicular bisectors meet.

12m

Enlargement with a positive scale factor

Multiply every distance from the centre by the scale factor. Sides grow by the same factor and angles stay the same.

12m

Enlargement with a fractional scale factor

A scale factor between 0 and 1 makes the image smaller and closer to the centre. It is still called an enlargement.

12m

Enlargement with a negative scale factor

A negative scale factor puts the image on the other side of the centre, upside down. Its size comes from the number without the sign.

12m

10.

Describing an enlargement: centre and scale factor

Join each image point back to its object point. The lines meet at the centre, and image length over object length is the scale factor.

12m

11.

Translation by a vector

A translation slides every point the same way. The column vector gives the steps right and the steps up.

12m

12.

Describing a single transformation fully

Name the transformation, then give everything it needs: mirror line; centre, angle and direction; centre and scale factor; or vector.

12m

13.

Combinations of transformations

Do the first transformation, then do the second to that image. Changing the order can change the answer.

13m

14.

Transformations: timed practice

A timed drill of mixed questions: name the transformation, find an image point, give the missing detail.

15.

Transformations: exam practice

Past-paper style questions on drawing and describing transformations, with full worked solutions and mark schemes.

15m

16.

Transformations: summary

The four transformations, what each description needs, and the common mistakes, on one page.

7.2 · Vectors in two dimensions

Column vectors and vector notation, adding and subtracting vectors, and multiplying a vector by a scalar.

0/9

Describing a translation using a column vector

A vector is a move with a size and a direction. The top number is the move right, the bottom number the move up.

Preview12m 2.

Vector notation: column vectors, AB and a

One move, three ways to write it. Equal vectors have the same size and direction, and BA is the negative of AB.

Preview12m

Adding vectors

Put the vectors end to start. The sum goes straight from the first start to the last end, and you add the top and bottom numbers.

12m

Subtracting vectors

Taking away b is the same as adding -b. Subtract the top numbers and subtract the bottom numbers.

12m

Multiplying a vector by a scalar

A scalar is a plain number. It multiplies both parts of the vector, and a negative scalar turns the vector round.

12m

Combining vector operations

Work out 2a + 3b one step at a time, and find the numbers p and q that make pa + qb equal a given vector.

13m

Vectors in two dimensions: timed practice

A timed drill of mixed questions: read a vector off a grid, add, subtract and multiply by a scalar.

Vectors in two dimensions: exam practice

Past-paper style questions on column vectors and vector arithmetic, with full worked solutions and mark schemes.

15m

Vectors in two dimensions: summary

Vector notation, the rules for adding, subtracting and scaling, and the common mistakes, on one page.

7.3 · Magnitude of a vector

The length of a vector from Pythagoras' theorem, modulus notation, exact answers, and the magnitude of a combined vector.

0/7

Calculating the magnitude of a vector

The two parts of a column vector are the two short sides of a right-angled triangle, so its length is the square root of x² + y².

Preview12m 2.

Modulus notation and exact answers

|a| means the magnitude of a. When the answer is not a whole number, leave it as a surd or round it to 3 significant figures.

Preview12m

The magnitude of a combined vector

Work out the vector 2a - b first, then find its magnitude. |a + b| is usually not |a| + |b|.

12m

Finding a missing component from the magnitude

If the magnitude is given, write x² + y² equal to its square and solve. There are often two answers.

12m

Magnitude of a vector: timed practice

A timed drill of magnitudes: whole-number answers, surds, rounded answers and combined vectors.

Magnitude of a vector: exam practice

Past-paper style questions on magnitude, with and without a calculator, with full worked solutions and mark schemes.

15m

Magnitude of a vector: summary

The magnitude formula, modulus notation, rounding rules and the common mistakes, on one page.

7.4 · Vector geometry

Directed line segments, position vectors, routes in terms of two vectors, and proofs about parallel lines, collinear points, ratio and similarity.

0/12

Representing vectors by directed line segments

An arrow shows a vector: its length is the magnitude and the arrowhead gives the direction. Equal arrows in a shape are equal vectors.

Preview12m 2.

Position vectors

The position vector of a point is the move from the origin O to that point. For A(1, 2) it is the column vector (1 2).

Preview12m

The vector between two points: AB = b - a

Go from A back to O, then from O to B. So AB = b - a: end minus start. The midpoint of AB has position vector (a + b)/2.

12m

Expressing vectors in terms of two coplanar vectors

Follow a route along known sides. Going against an arrow gives a minus sign. Collect the a terms and the b terms.

13m

Vectors in parallelograms and at midpoints

Opposite sides of a parallelogram are equal vectors. Halfway along a side is half of that vector.

13m

Showing that vectors are parallel

Two vectors are parallel when one is a number times the other. The number also tells you how many times longer it is.

12m

Showing that three points are collinear

Collinear means on one straight line. Show that AB is a multiple of AC, and say that they share the point A.

12m

Vector problems involving ratio

A point that splits AB in the ratio 2 : 3 is 2/5 of the way from A. So OP = a + 2/5 (b - a).

13m

Vector problems involving similarity

If XY = 1/3 AB, the lines are parallel and the small triangle is similar to the big one, with lengths one third as long.

13m

10.

Vector geometry: timed practice

A timed drill of mixed questions: routes, position vectors, parallel lines, collinear points and ratios.

10m

11.

Vector geometry: exam practice

Past-paper style vector geometry questions, including a full "show that" proof, with worked solutions and mark schemes.

15m

12.

Vector geometry: summary

Position vectors, routes, the parallel and collinear tests, ratio and the common mistakes, on one page.

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.