1. Chapter 3
Vectors
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Chap 3

2. 3.2 Vectors and Scalars
WHAT IS PHYSICS?
Physics deals with a great many quantities that have both size and direction, and it needs
a special mathematical language- the language of vectors- to describe those quantities.
This language is also used in engineering, the other sciences, and even in common speech. If
have ever given directions such as “Go five blocks down this street and then hang a left”
you have used the language of vectors. In fact, navigation of any sort is based on vectors,
but physics and engineering also need vectors in special ways to explain phenomena involving
rotation and magnetic forces, which we get to in later chapters. In this chapter, we focus on
the basic language of vectors.
Vectors
Quantities which indicate both magnitude and direction.
Examples: displacement, velocity, acceleration
Scalars
Quantities which indicate only magnitude
Examples: Time, speed, temperature, distance
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3. Arrows are used to represent vectors.
3.2 Vectors and Scalars
Arrows are used to represent vectors.
The length of the arrow signifies magnitude
The head of the arrow signifies direction
Sometimes the vectors are represented by bold lettering, such as vector a. Sometimes they are represented with arrows on the top, such as
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4. 3.3 Adding vectors geometrically
Vector a and vector b can be added geometrically to yield the resultant vector sum, s.
Place the second vector, b, with its tail touching the head of the first vector, a. The vector sum, s, is the vector joining the tail of a to the head of b. The vector s is the vector sum of vectors of a and b. The symbol + in the above equation and the words “sum” and “add” have different meanings for vectors than they do in the usual algebra because they involve both magnitude and direction.
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5. Some rules: 3.3 Adding vectors geometrically
Vector addition, defined in this way, has two important properties
First, the order of addition does not matter. Adding vector a to vector
b gives the same result as adding vector b to vector a.
When there are more than two vectors, we can group them
in any order as we add them. Thus, if we want to add
vectors a, b, and c, we can add vectors a and b first and
then add their vector sum to vector c. We can also add
vector b and c first and then add that sum to vector a. We get
the same result either way.
The vector –b is a vector with the same magnitude as vector b
but the opposite direction. Adding the two vectors would yield
zero.
We use this properties to define the difference between
two vectors.
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6. 3.3 Adding vectors…sample problem
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7. 3.4 Components of vectors
Adding vectors geometrically can be tedious. A neater and easier technique involves algebra but requires that the vectors be placed on a rectangular coordinate system.
The component of a vector along an axis is the projection of the vector onto that axis.
The process of finding the components of a vector is called resolution of the vector.
In 3-dimensions, there are three components of a vector along pre-defined x-, y-, and z-axes.
We can find the components of vector a geometrically
from the right triangle there :
ax = a cos θ and ay = a sin θ; where θ is the angle that
the vector a makes with the positive direction of the
x-axis, and a is the magnitude of vector ā.
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8. We find the components of a vector by using the right triangle rules.
3.4 Components of vectors
We find the components of a vector by using the right triangle rules.
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9. Example, vectors:
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10. 3.4: Problem Solving Check-points
Angles measured counterclockwise will be considered positive, and clockwise negative.
Change the units of the angles to be consistent.
Use definitions of trig functions and inverse trig functions to find components.
Check calculator results.
Check if the angles are measured counterclockwise from the positive direction of the x-axis, in which case the angles will be positive.
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11. 3.5: Unit vectors
A unit vector is a vector of unit magnitude, pointing in a particular direction.
Unit vectors pointing in the x-, y-, and z-axes are usually designated by
respectively
Therefore vector, ,
with components ax and ay in the x- and y-directions,
can be written in terms of the following vector sum:
Angles may be measured in degrees or radians (rad). To relate the two
measures, recall that a full circle is 3600 and 2π rad. To convert, say,
400 to radians, write
400 (2π rad)/3600 = 0.70 rad
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12. The procedure of adding vectors also applies to vector subtraction.
3.6: Adding vectors by components If
then
Therefore, two vectors must be equal if their corresponding components are equal.
The procedure of adding vectors also applies to vector subtraction.
Therefore,
where
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13. Example, vector addition:
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14. Example, vector addition:
Note:
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15. Freedom of choosing a coordinate system
3.7: Vectors and the Laws of Physics
Freedom of choosing a coordinate system
Relations among vectors do not depend on the origin or the orientation of the axes.
Relations in physics are also independent of the choice of the coordinate system.
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16. A. Multiplying a vector by a scalar
3.8: Multiplying vectors
A. Multiplying a vector by a scalar
Multiplying a vector by a scalar changes the magnitude but not the direction:
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17. B. Multiplying a vector by a vector: Scalar (Dot) Product
3.8: Multiplying vectors
B. Multiplying a vector by a vector: Scalar (Dot) Product
The scalar product between two vectors is written as:
It is defined as:
Here, a and b are the magnitudes of vectors a and b respectively, and f is the angle between the two vectors.
The right hand side is a scalar quantity.
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18. C. Multiplying a vector with a vector: Vector (Cross) Product
3.8: Multiplying vectors
C. Multiplying a vector with a vector: Vector (Cross) Product
The right-hand rule allows us to find the direction of vector c.
The vector product between two vectors a and b can be written as:
The result is a new vector c, which is:
Here a and b are the magnitudes of vectors a and b respectively, and f is the smaller of the two angles between a and b vectors.
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19. 3.8: Multiplying vectors; vector product in unit-vector notation:
Note that:
And,
j x k = i
i x k = j
i x j = k
ˆi .ˆi = ˆj .ˆj = kˆ . kˆ = 1
ˆi .ˆj = ˆj . kˆ = kˆ . ˆi = 0
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20. Sample problem, vector product
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21. Sample problem, vector product, unit vector notation
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