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Section 6.6
Vectors
Vectors involve both magnitude (length) and di ti
direction.
Example: You are driving due north at 50 mph.
The magnitude is 50 and the direction is due north.
Scalars involve only magnitude
Scalars involve only magnitude.
Example: The temperature outside is 88o . Directed line segments and geometric vectors
Q
P
•
initial point
•
terminal point
This is a directed line segment from P to g
JJJG
PQ JJJG
Q. Denote by .
The magnitude of is its length or PQ
JJJG
PQ
distance from P to Q. Denote by . Magnitude cannot be negative.
A vector is a directed line segment.
Denote vectors by boldface letters or with arrows over the letters. 1
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Two vectors are equal if they have the same magnitude and the same direction.
Vectors can be represented on the rectangular b
d
h
l
coordinate system. Vectors have a horizontal and a vertical component.
A vector starting at the origin and extending left 2 − 2, − 3
units and down 3 units is v = = ‐2i –
3j. Its magnitude is 2
2
v = (-2) + (−3) = 4 + 9 = 13
Sketch each vector as a position vector and find its magnitude.
v = 2i + 3jj
(0,0)
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v = ‐5j
Let v be the vector from initial point P1 to terminal point P2. Write v in terms of i and j.
P1 = (2, ‐5), P
( , ), 2 = (‐6, 6)
( , )
Use v = (x2 – x1)i + (y2 – y1)j
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Operations with Vectors in Terms of i and j.
Adding and Subtracting Vectors in Terms of i and j.
If v
f = a1i + b
b1j and w
d = a2i + b
b2j, then
h
v + w = (a1 + a2)i + (b1 + b2)j
v – w = (a1 – a2)i – (b1 – b2)j
Scalar Multiplication
If v = ai + bj and k is a real number, then the scalar multiplication of the vector v and the scalar k is kv = (ka)i + (kb)j
Given u = 2i – 5j, v = ‐3i + 7j, and w = ‐I ‐6j
Find:
v + w
v‐w
6v
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4w – 3v
u-w
Let u = ‐2i + 3j, v = 6i – j, w = ‐3i. Find each specified vector or scalar.
3u – (4v – w)
v+w − v-w
2
2
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Zero Vector
The vector whose magnitude is 0. It is assigned no direction. It can be expressed in terms of i and j using 0 = 0i + 0j.
Unit Vector
A vector whose magnitude is 1.
Find the unit vector that has the same direction as the vector v.
v = ‐5j
v = 8i – 6j
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Write the vector v
Write
the vector v in terms of i
in terms of i and j
and j whose magnitude||v|| whose magnitude||v||
and direction angle θ are given.
||v|| = 8, θ = 45o
||v|| = ¼, θ
||v|| ¼, θ = 200
200o
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In each problem below, a vector is described. Express the vector in terms of i and j. If exact values are not possible, round components to the nearest tenth.
A child pulls a sled along level ground by exerting a force of 30 pounds A
child pulls a sled along level ground by exerting a force of 30 pounds
on a handle that makes an angle of 45o with the ground.
A plane with an airspeed of 450 miles per hour is flying in the direction N35oW.
y
F1
F2
x
A force vector is one that represents a pull or push of some type. If F1 and F2 are two forces simultaneously acting on an object, the vector sum F1 + F2 is the resultant force.
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The magnitude and direction exerted by two tugboats towing a ship are 4200 pounds, N65oE, and 3000 pounds, S58oE, respectively. Find the magnitude, to the nearest pound, and direction angle, to the nearest tenth of a degree, of the resultant force.
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