VECTORS IN MECHANICS

Magnitude and DirectionIf a vector, a, is given in component form e.g. a = 4i + 3j, wecan use Pythagoras’ theorem to find the magnitude, and basic trigonometry to find the direction.

4

3a

θ~a

Note, when printed, avector is written in boldprint. When handwritten,vectors are underlined:

The magnitude of a is written as either or just: a

It would be handwritten as:

a

a~ or a

Here we have: 22 3 4 = 5

The direction to the horizontal is θ, where tan θ = 34

θ = 36.9º

=a

Example 1: Given the vectors a = 3i + j, and b = 6i – 7j, find the magnitude and direction of i) 2a + b ii) a – b. Givethe directions as three figure bearings.

a =

1

3b =

6

7

i) 2a + b =

1

32 +

6

7

=12

5

12

5

θ

2a + b = 22 5 12 = 13

tan θ = 512

22.6º

The bearing is 90 + 22.6 = 113º ( to the nearest degree).

θ =

Note, the vector pi + qj can be written as a column vector

q

p

a =

1

3b =

6

7

ii)

3

8

i

j

θ3

8

a – b = 8.54 (3 s.f.)

tan θ =83

69.4º

The bearing is 270 + 69.4 = 339º (to the nearest degree).

a – b =

θ =

22 8 3 =

Finding the components of a vector:

Consider a vector of magnitude 25which makes an angle of 40º to thehorizontal.

i

j

40º

25

The horizontal component can be found:

The vertical component can be found:

cos 40 =adjacent 25

sin 40 =opposite 25

Hence the vector is:

16.1

19.2

40sin 25

40 cos 25= (or 19.2i + 16.1j).

adjacent = 25 cos 40

opposite = 25 sin 40

Example 2: A vector of magnitude 17 is shown. Find the vector in component form.

i

j

17

23º

17

23º

cos 23 =adjacent 17

sin 23 =opposite 17

Hence the vector is:15.6

6.64

17 cos 23

17 sin 23

=

opposite = 17 sin 23

adjacent = 17 cos 23 Note the negativefor both of thecomponents.

Example 3: A vector of magnitude 35 is shown. Find the vector in component form.

i

j

51º

35

51º35

cos 51 =adjacent 35

sin 51 =opposite 35

Hence the vector is:27.2

22.0

35 sin 51

35 cos 51

=

adjacent = 35 cos 51

opposite = 35 sin 51

Summary of key points:

This PowerPoint produced by R.Collins; ©ZigZag Education 2010

If a vector is given in component form, Pythagoras’ theorem canbe used to find the magnitude, and basic trigonometry to find thedirection.

θk

h R

The component of a vector oppositea given angle is found using sin.

The component of a vector adjacentto a given angle is found using cos.

i.e. h = R sin θ

i.e. k = R cos θ