1. Chapter 2 : Kinematics
โข Objectives :
2-1 Scalars versus Vectors
2-2 Components of a Vector
2-3 Adding and Subtracting Vectors
2-4 Unit vectors
2-5 Position, Displacement, Velocity and Acceleration Vectors
2. 2-1 Scalars versus Vectors
โข A scalar is a number with units. A vector on the other hand is
a mathematical quantity with both a direction and a
magnitude
โข A vector is indicated with a boldface or written with a small
arrow above it. N
E
๐
๐
๐
Displacement vector
3. 2-2 Components of a Vector
โข To find the components of a vector we need to set up a
coordinate system.
โข In 2d we choose an origin, O and a positive direction for both
the x and y axes. (If 3d system we would also indicate a z axis)
โข A vector is defined by its magnitude (indicated by the length
of the arrow representing the vector)
and its direction.
โข The quantities of ๐๐ฅ and ๐๐ฆ are referred
to as the x and y scalar components
of the vector ๐
y
x
๐
๐๐ฅ
๐๐ฆ
4. 2-2 Components of a Vector
โข We can find the components of a vector by using standard
trigonometric relations, as summarized below.
y
x
๐ด
๐ด๐ฅ
๐ด๐ฆ
๐
๐ด๐ฅ = ๐ด cos ๐
๐ด๐ฆ = ๐ด sin ๐
๐ด = ๐ด๐ฅ
2
+ ๐ด๐ฆ
2
๐ = ๐ก๐๐โ1
๐ด๐ฆ
๐ด๐ฅ
๐๐๐๐๐๐ก๐ข๐๐ ๐๐ ๐๐๐๐ก๐๐๐ด
๐๐๐๐๐๐ก๐๐๐ ๐๐ ๐๐๐๐ก๐๐๐ด
๐ฅ ๐ ๐๐๐๐๐ ๐๐๐๐. ๐๐ ๐๐๐๐ก๐๐๐ด
y ๐ ๐๐๐๐๐ ๐๐๐๐. ๐๐ ๐๐๐๐ก๐๐๐ด
5. 2-2 Components of a Vector
โข How do you determine the correct signs for the x and y
components of a vector?
โข To determine the signs of a vectorโs components, it is only
necessary to observe the direction in which they point.
โข For ex. In the case of a vector in the 2nd quadrant of
the coordinate system, the x component
points in the negative direction; hence
๐ด๐ฅ < 0 and the y component points
in the positive direction hence ๐ด๐ฅ < 0.
Here the direction is 180ยฐ โ ๐
y
x
๐ด
๐ด๐ฅ
๐ด๐ฆ
๐
๐ด
๐ด๐ฅ
๐ด๐ฆ
๐
๐ด
๐ด๐ฆ
๐
๐ด
๐ด๐ฆ
๐
6. 2-3 Adding and Subtracting Vectors
In this section we begin by defining vector addition graphically,
and then show how the same addition can be performed more
concisely using and accurately using components.
2 Ways to do vector addition:
โข Adding vectors graphically
โข Adding vectors using components
7. 2-3 Adding and Subtracting Vectors
2.3.1 Adding vectors graphically
โข It can be done using either
- Triangle method ( for 2 vectors)
- Parallelogram method (for 2 vectors)
- Vector polygon method (for >2 vectors)
โข Vector ๐ถ is the vector sum of ๐ด and ๐ต
โข To add the vectors, place the tail of ๐ต at the head
of ๐ด. The sum, ๐ถ = ๐ด + ๐ต, is the vector extending from the
tail of ๐ด to the head of ๐ต
โข Itโs fine to move the arrows, as long as you donโt change their
length or their direction. After all, a vector is defined by its length
and direction, if these are unchanged, so is the vector.
๐ด
๐ต
๐ถ
๐ถ = ๐ด + ๐ต
๐ฆ
๐ฅ
8. 2-3 Adding and Subtracting Vectors
2.3.2 Adding vectors using components
โข The graphical way of adding vectors yields approximate
results, limited by the accuracy with which the vectors can be
drawn and measured.
โข In contrast, exact results can be obtained by adding the
vectors in terms of their components.
๐ด
๐ต
๐ฆ
๐ฅ
๐ถ = ๐ด + ๐ต
๐ด
๐ต
๐ฆ
๐ฅ
๐ถ
๐ด๐ฅ
๐ด๐ฆ
๐ต๐ฅ
๐ต๐ฆ
๐ถ๐ฅ = ๐ด๐ฅ+ ๐ต๐ฅ
๐ถ๐ฆ = ๐ด๐ฆ+ ๐ต๐ฆ
๐ถ = ๐ถ๐ฅ
2
+ ๐ถ๐ฆ
2
๐ = ๐ก๐๐โ1
๐ถ๐ฆ
๐ถ๐ฅ
๐ฆ
9. 2-3 Adding and Subtracting Vectors
2.3.3 Subtracting vectors
โข The negative of a vector is
represented by an arrow of the
same length as the original vector,
but pointing in the opposite direction.
โข That is, multiplying a vector by minus one reverses its
direction
๐ถ = ๐ด โ ๐ต
๐ด
๐ต
๐ต
๐ถ
๐ถ = ๐ด + (โ๐ต)
๐ฆ
๐ฅ
10. 2-4 Unit Vectors
โข The unit vectors of an x-y coordinate system, ๐ฅ ๐๐๐ ๐ฆ, are
defined to be dimensionless vectors of unit magnitude 1,
pointing in the positive x and y directions.
โข It is used to provide a convenient way of expressing an
arbitrary vector in terms of its components
y
x
๐ฅ
๐ฆ
11. 2-4 Unit Vectors
Multiplying Unit Vectors by Scalars
โข This will increase its magnitude by a factor of x, but does not
change its direction. For example magnitude x = 3
โข Now we can write a vector ๐ด in terms of its x and y vector
components
๐ด = ๐ด๐ฅ๐ฅ + ๐ด๐ฆ๐ฆ
y
x
๐ด 3๐ด
โ3๐ด
12. 2-4 Unit Vectors
Multiplying Unit Vectors by Scalars
โข Now we can write a vector ๐ด in terms of its x and y vector
components
๐ด = ๐ด๐ฅ๐ฅ + ๐ด๐ฆ๐ฆ
โข We see that the vector components are the projection of a
vector onto the x and y axes.
โข The sign of the vector components is positive if they point in
the positive x or y direction and vice versa.
y
x
๐ด ๐ด๐ฆ๐ฆ
๐ด๐ฅ๐ฅ
13. 2-5 Position, Displacement, Velocity
and Acceleration Vectors
โข Position vectors
โข Velocity vectors
โข Acceleration Vectors