Teach A Level Maths Vectors for Mechanics

Volume 4: Mechanics 1 Vectors for Mechanics

Many of the quantities in Mechanics are vectors. This presentation covers the vector theory that you need in M1. The theory is in sections so that, if you wish, you can do the parts as you need them. 1. Introducing Vectors 3. Position Vectors and Displacement 2. The Unit Vectors and i j 4. Column Vectors

Many of the quantities in Mechanics are vectors. This presentation covers the vector theory that you need in M1. The theory is in sections so that, if you wish, you can do the parts as you need them. 1. Introducing Vectors 3. Position Vectors and Displacement 2. The Unit Vectors and i j 4. Column Vectors Click on the section you want.

1. Introducing Vectors

A vector can be shown by a line segment with an arrow. This vector is written as AB A B A B BAAB  The arrow for runs from B to A. BA A B AB

O The magnitude ( size ) of a vector is shown by the length of the line. The grid has 1 cm squares. PQ AB has magnitude 3. We write | AB |  3. B The magnitude of... PQ is given by | PQ |  4. A

R O S We can use Pythagoras’ theorem to find the magnitude of other vectors. Using Pythagoras’ theorem, RS 2   13  RS  3·61 ( 3 s.f. )

e.g. AB  A B is equivalent to the sum of any vectors starting at A and ending at B. AB P PBPBAPAP  Notice the directions of the arrows on the vectors. When we draw a vector and when we write it, the “head” always points towards the 2 nd letter. head of tail of AP PB and are drawn head-to-tail. They are added to give. APPB AB

EXERCISE (i) B (ii) (v) A (iv) (vi) B C (iii) Answers: 1(a) (v) (b) (ii) (c) (iv) 2.What is the magnitude of (a) (b) (c)  AB BC AB BC 1.Which vector in the diagram is equal to (a) (b) BA   (c) AB BC (d) BC  (d) (i) 2(a) 4 (b) 3 (c) 5

The next section looks at Unit Vectors. Choose an option below. Vector Menu Return to Previous Presentation Continue to Unit Vectors

2. The Unit Vectors and i j

j i Instead of drawing diagrams to show vectors we can use unit vectors. They have magnitude 1. e.g. A velocity v is given by v  3  4 i j x y j i 3 4 v The unit vectors and are parallel to the x- and y- axes respectively. ij In text-books single letters for vectors are printed in bold but we must underline them.

v  3  4 i j x y 3 4 v |v|  3 2  4 2 j i No or in magnitude i j |v|  3 2  4 2 So, if we have the unit vector form, we use the numbers in front of and i j |v|  5  Tip: Squares of real numbers are always positive so we never need any minus signs. We can write |v| for speed. The magnitude of velocity is speed, so, using Pythagoras’ theorem,

x y 3 4 v j i v   i j The direction of the vector is found by using trig.  tan   53·1  ( 3 s.f. )  BUT beware ! 3443

If we need the direction of a vector when unit vectors are used, we must sketch the vector to show the angle we have found. v  3  4 i j Suppose Without a diagram we get tan  33 44  53·1  ( 3 s.f. ) So again But, the vectors are not the same ! v   i j  53·1  ( 3 s.f. ) 3443 For we have 3 4  i j 3443 v   3 4 v  3  4 i j 

(a) magnitude 2 due east (b) magnitude 5 due south Ans: 2 i Ans:  5 j Any vector parallel to and can easily be written in the and form. i j i j Tell your partner what the following vectors would be in and form where and are unit vectors due east and due north respectively. i j i j j i We will see how to deal with vectors in other directions later.

The next 2 pages are needed only by those of you taking the MEI/OCR specification. SKIP to next Vector Section CONTINUE Other Vector Options Return to Previous Presentation

Using and we can describe motion in 2 -dimensions. i j If we want to predict the positions of two aircraft, for example, to make sure that they are not about to collide, we need 3 -D. j The method used to write a quantity in 3 -D is just an extension of that for 2 -D. i k x y z

e.g. The velocity of an aircraft is given by v  3  4  2 i j k Find the speed. Solution: The magnitude of the vector gives the speed. 3 2  42 42 2 ( You won’t be asked to find the direction.) No minus sign  v  29  v  v  5·39 ( 3 s.f. )

The next section looks at Position Vectors and Displacement. Choose an option below. Other Vector Options Return to Previous Presentation Continue to Position Vectors and Displacement

3. Position Vectors and Displacement

But, AO OB  AB If a body moves from A to B, gives the displacement of B from A. AB x y O B A We often use single letters for position vectors, so a  OB OA and b  and are called the position vectors of A and B. OAOB A position vector gives the position of a point relative to the origin. b a  a  b aa AB  b  a  So, AB 

Notation: The letter r is often used for the position vector of a body. Rearranging to find the position vector of B gives AB  b  a So, may be given as s  r B  r A Since displacement is given by s, we can have r B  r A  s x y O B A r Ar A s r Br B or x y O B A b a AB

Solution: r A  i  3  2 ) km j Find the position vector of B. y x O B A s r B  r A  s The displacement of B from A is  s  AB 6 j  r B  i  3  2 ) j  6 j  r B   3  4 ) km i j e.g.The position vector of a point A is given by A body moves from A to a point B where B is 6 km due north of A. 6 km r Br B r Ar A In the following, and are unit vectors east and north respectively. i j

r A  i  6  3 ) m j The position vector of a point A is given by A body moves from A to a point B where B is 5 m due west of A. Find EXERCISE (a) the displacement of B from A, and (b) the position vector of B. Solution:  s  AB (a) r B  r A  s  r B  i  6  3 ) j  r B   3 ) m i j (b)  5 m i  5 i

The next section looks at Column Vectors. Choose an option below. Other Vector Options Return to Previous Presentation Continue to Column Vectors

4. Column Vectors

Column vectors are just vectors written in columns ! e.g. The position vectors of A and B are given by a    6 i j b  and i 4  2 j Solution: a  4 22 b  6 11 and  b  a AB  6 11 4 22     Write a and b as column vectors and find. AB

Column vectors are just vectors written in columns ! e.g. The position vectors of A and B are given by a    6 i j b  and i 4  2 j Solution: a  4 22 b  6 11 and  b  a AB   6 11 4 22   1  4 AB   Write a and b as column vectors and find. AB

Column vectors are just vectors written in columns ! e.g. The position vectors of A and B are given by a    6 i j b  and i 4  2 j Solution: a  4 22 b  6 11 and  b  a AB   6 11 4 22  6   2  1  4 AB    Write a and b as column vectors and find. AB

Column vectors are just vectors written in columns ! e.g. The position vectors of A and B are given by a    6 i j b  and i 4  2 j Solution: a  Write a and b as column vectors and find. AB 4 22 b  6 11 and  b  a AB   6 11 4 22  6   2  1  4 AB   8 55  Tip: You don’t have to convert from and form to column vectors as either form can be used. However, students make fewer sign errors using column vectors. i j

Choose an option below. Repeat a Vector Section Return to Previous Presentation This is the end of the final section of Vectors for Mechanics.