# Vectors by IbiOrigin

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Vectors
Lecture Outline
•   Vectors and Scalars
•   Presentation of Vectors
•   Addition and Subtraction of vector
•   Component of Vector
Vectors and Scalars

•A vector has magnitude as well as direction.
•Some vector quantities: displacement, velocity,
force, momentum
•A scalar has only a magnitude.
•Some scalar quantities: mass, time,
temperature
Presentation of Vectors
• On a diagram, each vector
is represented by an arrow
• Arrow pointing in the
direction of the vector
• Length of arrow is
proportional to the
magnitude of the vector

• Symbol V or V
• Magnitude of the vector:

V or V

For vectors in one
dimension, simple
are all that is needed.
You do need to be
as the figure indicates.
If the motion is in two dimensions, the situation is
somewhat more complicated.
Here, the actual travel paths are at right angles to one
another; we can find the displacement by using the
Pythagorean Theorem.
Adding the vectors in the opposite order gives the
same result:
Even if the vectors are not at right angles,
they can be added graphically by using the
tail-to-tip method.
The parallelogram method may also be used; here
again the vectors must be tail-to-tip.
Subtraction of Vectors

In order to subtract vectors, we define
the negative of a vector, which has the
same magnitude but points in the
opposite direction.

Then we add the negative vector.
Multiplication of a Vector by a Scalar

A vector V can be multiplied by a scalar c;

the result is a vector c V that has the same
direction but a magnitude cV. If c is
negative, the resultant vector points in the
opposite direction.
ConcepTest 3.1a               Vectors I
1) same magnitude, but can be in any
If two vectors are given
direction
such that A + B = 0, what 2) same magnitude, but must be in the same
can you say about the     direction
magnitude and direction    3) different magnitudes, but must be in the
same direction
of vectors A and B?
4) same magnitude, but must be in opposite
directions
5) different magnitudes, but must be in
opposite directions
ConcepTest 3.1a                Vectors I
1) same magnitude, but can be in any
If two vectors are given
direction
such that A + B = 0, what 2) same magnitude, but must be in the same
can you say about the     direction
magnitude and direction    3) different magnitudes, but must be in the
same direction
of vectors A and B?
4) same magnitude, but must be in opposite
directions
5) different magnitudes, but must be in
opposite directions

The magnitudes must be the same, but one vector must be pointing in
the opposite direction of the other in order for the sum to come out to
zero. You can prove this with the tip-to-tail method.
ConcepTest 3.1b                  Vectors II

Given that A + B = C, and     1) they are perpendicular to each other
that lAl 2 + lBl 2 = lCl 2,   2) they are parallel and in the same direction
how are vectors A and B       3) they are parallel but in the opposite
oriented with respect to      direction
each other?
4) they are at 45° to each other
5) they can be at any angle to each other
ConcepTest 3.1b                  Vectors II

Given that A + B = C, and     1) they are perpendicular to each other
that lAl 2 + lBl 2 = lCl 2,   2) they are parallel and in the same direction
how are vectors A and B       3) they are parallel but in the opposite
oriented with respect to      direction
each other?
4) they are at 45° to each other
5) they can be at any angle to each other

Note that the magnitudes of the vectors satisfy the Pythagorean
Theorem. This suggests that they form a right triangle, with vector C
as the hypotenuse. Thus, A and B are the legs of the right triangle and
are therefore perpendicular.
ConcepTest 3.1c                 Vectors III
Given that A + B = C,   1) they are perpendicular to each other
and that lAl + lBl =    2) they are parallel and in the same direction
lCl , how are vectors   3) they are parallel but in the opposite direction
A and B oriented with
4) they are at 45° to each other
respect to each
other?                  5) they can be at any angle to each other
ConcepTest 3.1c                Vectors III
Given that A + B = C,   1) they are perpendicular to each other
and that lAl + lBl =    2) they are parallel and in the same direction
lCl , how are vectors   3) they are parallel but in the opposite direction
A and B oriented with
4) they are at 45° to each other
respect to each
other?                  5) they can be at any angle to each other

The only time vector magnitudes will simply add together is when the
direction does not have to be taken into account (i.e., the direction is
the same for both vectors). In that case, there is no angle between
them to worry about, so vectors A and B must be pointing in the
same direction.
Any vector can be expressed as the sum of two
other vectors, which are called its
components. The process of finding the
component is known as resolving the vector
into its component.
Because x and y axis is
perpendicular, they can be
calculate using trigonometric
functions.
The components are effectively one-dimensional, so
1. Draw a diagram
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use:

and                  .
Example 3-2: Mail carrier’s
displacement.
A rural mail carrier leaves the
post office and drives 22.0 km in
a northerly direction. She then
drives in a direction 60.0° south
of east for 47.0 km. What is her
displacement from the post
office?
Example 3-3: Three short trips.
An airplane trip involves three
legs, with two stopovers. The first
leg is due east for 620 km; the
second leg is southeast (45°) for
440 km; and the third leg is at
53° south of west, for 550 km, as
shown. What is the plane’s total
displacement?
ConcepTest 3.2             Vector Components I

1) it doubles
If each component of a
2) it increases, but by less than double
vector is doubled, what
3) it does not change
happens to the angle of
4) it is reduced by half
that vector?              5) it decreases, but not as much as half
ConcepTest 3.2                Vector Components I

1) it doubles
If each component of a
2) it increases, but by less than double
vector is doubled, what
3) it does not change
happens to the angle of
4) it is reduced by half
that vector?                 5) it decreases, but not as much as half

The magnitude of the vector clearly doubles if each of its
components is doubled. But the angle of the vector is given by tan
q = 2y/2x, which is the same as tan q = y/x (the original angle).

Follow-up: If you double one component and not
the other, how would the angle change?

You are adding vectors of length          1) 0

20 and 40 units. What is the only         2) 18

possible resultant magnitude that         3) 37

you can obtain out of the                 4) 64

following choices?                        5) 100

You are adding vectors of length          1) 0

20 and 40 units. What is the only         2) 18

possible resultant magnitude that         3) 37

you can obtain out of the                 4) 64

following choices?                        5) 100

The minimum resultant occurs when the vectors
are opposite, giving 20 units. The maximum
resultant occurs when the vectors are aligned,
giving 60 units. Anything in between is also
possible for angles between 0° and 180°.

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