1. VECTORS IN MECHANICS

2. Magnitude and Direction
If a vector, a, is given in component form e.g. a = 4i + 3j, we can use Pythagoras’ theorem to find the magnitude, and basic trigonometry to find the direction. 4 3 a θ ~ a
Note, when printed, a
vector is written in bold
print. When handwritten,
vectors are underlined:
The magnitude of a is written as either a
or just: a
It would be handwritten as: a ~
or a
Here we have: = a = 5 3 4
The direction to the horizontal is θ, where tan θ =
θ =
36.9º

3. Example 1: Given the vectors a = 3i + j, and b = 6i – 7j, find the
magnitude and direction of i) 2a + b ii) a – b. Give
the directions as three figure bearings.
Note, the vector pi + qj can be written as a column vector
a =
b = 2 + = i)
2a + b = 12 5 θ
2a + b =
= 13 5 12
tan θ =
θ =
22.6º
The bearing is =
113º
( to the nearest degree).

4. (to the nearest degree).
ii)
a =
b = i j θ 3 8
a – b =
a – b = =
8.54
(3 s.f.) 8 3
tan θ =
θ =
69.4º
The bearing is =
339º
(to the nearest degree).

5. Finding the components of a vector:j
40º 25
Consider a vector of magnitude 25
which makes an angle of 40º to the
horizontal.
adjacent 25
The horizontal component can be found:
cos 40 =
adjacent = 25 cos 40
opposite 25
The vertical component can be found:
sin 40 =
opposite = 25 sin 40 =
Hence the vector is:
(or 19.2i j).

6. Example 2: A vector of magnitude 17 is shown. Find the vector in
component form. i j 17
23º 17
23º
adjacent 17
cos 23 =
adjacent = 17 cos 23
Note the negative
for both of the
components.
opposite 17
sin 23 =
opposite = 17 sin 23 =
Hence the vector is:

7. Example 3: A vector of magnitude 35 is shown. Find the vector in
component form. i j
51º 35
51º 35
opposite 35
sin 51 =
opposite = 35 sin 51
adjacent 35
cos 51 =
adjacent = 35 cos 51 =
Hence the vector is:

8. If a vector is given in component form, Pythagoras’ theorem can
Summary of key points:
If a vector is given in component form, Pythagoras’ theorem can
be used to find the magnitude, and basic trigonometry to find the
direction. θ k h R
The component of a vector opposite
a given angle is found using sin.
i.e. h = R sin θ
The component of a vector adjacent
to a given angle is found using cos.
i.e. k = R cos θ
This PowerPoint produced by R.Collins; ©ZigZag Education 2010