Introduction to Vectors
Vectors help us to describe the motion of objects that do
not travel along a straight line.
Mathematically we describe motion in one dimension (or along a
straight line) as positive and negative(aka forward and
backwards).
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Vectors indicate direction; scalars do not
A scalar is a quantity that has magnitude (value) but not direction.
Examples:
• speed
• volume
• number of pages in a textbook
A vector is a physical quantity that has both direction and
magnitude
Examples:
• displacement
• velocity
• acceleration
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Notation:
Vector quantities are indicated by either boldface type or a
letter with an arrow over it.
Example:
Velocity:
v or v
Graphically vectors are shown as arrows that point in the
direction of the vector.
The length of a vector arrow in a diagram is proportional
to the vector's magnitude.
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A quantity with magnitude and direction is represented by a directed line segment PQ with initial point P and terminal point Q.
Let u be represented by the directed line segment from P = (0, 0) to Q = (3, 1), and let v be represented by the directed line segment from R = (2, 2) to S = (5, 3). Show that u = v.
S(5, 3)
It's magnitude (or length) is denoted by ||PQ|| and can be found by using the distance formula
R(2, 2)
v
Q(3, 1)
u
The vector v = PQ is the set of all directed line segments of length ||PQ|| which are parallel to PQ.
Two vectors, u and v, are equal if the line segments representing them are parallel and have the same length or magnitude.
v
u
Since PQ and RS have the same length they have the same magnitude.
P(0, 0)
Also, both line segments have the same direction because they are both directed toward the upper right on lines having slopes of 1/3.
Since PQ and RS have the same magnitude and direction, u = v.
Directed Line Segment
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Unit Vector
Has a magnitude of 1.
||v|| = 1
Zero Vector
Has magnitude of 0.
Vector Operations
||v|| = 0
Scalar multiplication is the product of a scalar, or real number, times a vector.
For example, the scalar 3 times v results in the vector 3v, three times as long and in the same direction as v.
v 3v
The product of ­ and v gives a vector half as long as and in the opposite direction to v. Aug 13­3:50 PM
v
­ v
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Vector Addition
The answer found by adding two vectors is called
the resultant vector.
When adding vectors, be sure they have the same units and
describe similar quantities.
ex: you can not add meters and feet together
v
To add vectors u and v: u
1. Place the initial point of v at the terminal point of u. be sure when you do this that you don't change the vector's magnitude or direction!
u + v
2. Draw the vector with the same initial point as u and
the same terminal point as v. This sum is called the resultant vector.
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Adding Vectors Graphically
Vector Subtraction
The length of the
resultant is the
magnitude. The
direction can be
found by using a
protractor.
Magnitude and direction from your
friend's house to the mall.
resultant
v
To subtract vectors u and v: u
1. Place the initial point of v at the initial point of u. again...be sure when you do this that you don't change the vector's magnitude or direction!
Magnitude and direction from your
house to your friend's house.
u − v 2. Draw the vector u − v from the terminal point of v to the terminal point of u.
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Examples: Given vectors u = (4, 2) and v = (2, 5)
Examples: Given vectors u = (1, 2) and v = (3, 1)
y
­2u = ­2(4, 2)
(4, 2)
u
(­8, ­4)
u + v = (4, 2) + (2, 5)
y
(2, 5)
v
u + v
u(4, 2)
x
y
u + v = (1, 2) + (3, 1) x
(1, 2)
­2u
u − v = (4, 2) − (2, 5)
y
(2, 5)
v
(4, 2)
u
x
u − v
u
u − v = (1, 2) − (3, 1) 2u − 3v = 2(1, 2) − 3(3, 1) = (­7, 1)
y
y
(1, 2)
u
(3, 1)
v
u + v
(3, 1)
v
x
x
2(1, 2)
u
3(3, 1)
v
x
Examples: Operations on Vectors Aug 17­9:43 AM
Do pgs. 456 & 457
#'s 15 ­ 24
Due Tomorrow.
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A car travels 20.0 km due north and then 35.0 km in a direction
60o west of north. Find the magnitude and direction of the
car's resultant displacement.
35
20
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Components of a Vector
•
•
•
•
Find the components of the velocity of a helicopter traveling 95 km/hr
at an angle of 35o to the ground.
Projections of vectors along coordinate axes.
The x component is parallel to the x-axis
The y component is parallel to the y-axis
You can often describe an object's motion more conveniently
by breaking a single vector into two components, or resolving the vector.
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A hiker walks 27.0 km from her base camp at 35o south of east.
The next day, she walks 41.0 km in a direction 65o north of east and
discovers a forest ranger's tower.
Find the magnitude and direction of her resultant displacement
between the base camp and the tower.
Homework:
Pgs. 457-459, #'s 71-73, 75, 77, 78, 84 & 85
Due tomorrow.
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