We look at a general vector , see Fig. 3.11 , which is decomposed into the sum of two vectors along the and axes. We define as a unit vector in the -direction, and as a unit vector in -direction. So , . Thus
where and are the components of in the and directions. The vecor as represented by the vector is called a coordinate vector.
As shown in Fig. 3.12 , the result in three dimensions is quite similar. Let , , be the right-handed set of unit vectors in the direction, respectively. [A set of vectors is called right-handed if, when turning a corkscrew from the first to the second vector, it moves in the direction of the third.] Thus
where , and are the components of .
We shall often use the notation for a vector . Once again the vectors and were given as position vectors, the displacement vector for the point starting from the origin. Using pythagoras’ theorem repeatedly we see that , and thus .
Let
then
(please verify these geometrically for 2 dimensional space)
Example 3.1:
Given the points and , find the component from for the vector .
Solution:
We realise that , or, . We thus find that .
We study or , .Then the unit vector in the direction of is
Clearly .
Example 3.2:
If find , and the direction cosines (dc’s) of .
Solution:
The d.c’s are the components of , i.e., , , .
If , and , and and are scalars, then
Example 3.3:
If , , find (i) , (ii) , (iii) , and (iv) the unit vector in the direction of .
Solution:
(i) , (ii) , (iii) , (iv)
Example 3.4:
Given the points
and
find: (i) The position vectors of A and B
relative to the origin
(ii) the vector
,
(iii) the position vector of the
mid-point
of
.
Solution:
(i)
,
.
(ii)
or
(iii)
Example 3.5:
A truck of mass 10000 kg stands on a
slope that makes and angle of
with the horizontal.
1) Find the acceleration of the truck
in terms of
.
2) An explosion exerts a force
orthogonal to the surface. Find the resultant force
(use
).
Solution:
1) Look at Fig.
3.13
. We see that the force
parallel to the plane is
, orthogonal
. The acceleration is thus
.
2) The new force, choosing the
axis parallel to the slope, and
orthogonal (upwards), is
. This has size
, and makes an angle of
with the slope, so
with the horizontal.