Vectors
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Vectors and Scalars
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Properties of Vectors
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Components of a Vector and Unit Vectors
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Homework
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Vectors and Scalars
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Vector - quantity that has magnitude and direction
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e.g. displacement, velocity, acceleration, force
Scalar - quantity that has only magnitude
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e.g. Time, mass, energy
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Displacement Vector
As a particle moves from A to B along the path represented by the dashed curve, its displacement is the vector
shown by the arrow from A to B.
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Adding Vectors
When vector B is added to vector A, the resultant R is the
vector that runs from the tail of A to the head of B.
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Commutative Property of Vector Addition
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The vector R resulting from the addition of the vectors
A and B is the diagonal of a parallelogram of sides A
and B.
Vector addition is commutative, that is A + B = B + A.
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Associative Property of Vector Addition
A+(B+C) = (A+B)+C
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Subtraction of Vectors
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To subtract vector B from vector A, simply add the
vector -B to vector A.
The vector -B is equal in magnitude and opposite in
direction to the vector B.
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Components of a Vector
A vector A lying in the xy plane can be represented by its
component vectors Ax and Ay .
Ax = A cos θ
A=
s
A2x
+
Ay = A sin θ
Ay
tan θ =
Ax
A2y
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Unit Vectors
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The unit vectors i, j, and k are directed along the x, y,
and z axes, respectively.
The unit vectors i, j, and k form a set of mutually perpendicular vectors and the magnitude of each unit
vector is one
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|i|=|j|=|k|=1
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Vectors in Component Form
A vector A lying in the xy plane has component vectors
Axi and Ay j where Ax and Ay are the components of A.
A=Axi + Ay j
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Example 1
A small plane leaves an airport on an overcast day and
later is sighted 215 km away, in a direction making an
angle of 22◦ east of north. (a) How far east and north
is the airplane from the airport when sighted? (b) Using
a coordinate system with the y-axis pointing north and
the x-axis east, write the position of the airplane in unit
vector notation.
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Example 1 Solution
A small plane leaves an airport on an overcast day and
later is sighted 215 km away, in a direction making an
angle of 22◦ east of north. (a) How far east and north is
the airplane from the airport when sighted?
N
y
ry r
θ
rx
x
θ = 90◦ − 22◦ = 68◦
rx = r cos θ = (215 km) cos 68◦ = 81 km
ry = r sin θ = (215 km) sin 68◦ = 199 km
(b) Using a coordinate system with the y-axis pointing
north and the x-axis east, write the position of the airplane in unit vector notation.
r = rxi + ry j = (81 km) i + (199 km) j
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Vector Addition Using Components
R=A+B
Rxi + Ry j = (Axi + Ay j) + (Bxi + By j)
Rxi + Ry j = (Ax + Bx) i + (Ay + By ) j
Rx = A x + B x
Ry = A y + B y
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Example 2
Find R = A + B + C where A = 4.2i - 1.6j, B = -3.6i + 2.9j,
and C = -3.7j.
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Example 2 Solution
Find R = A + B + C where A = 4.2i - 1.6j, B = -3.6i + 2.9j,
and C = -3.7j.
R = Rx i + R y j
R = (Ax + Bx + Cx) i + (Ay + By + Cy ) j
R = (4.2 − 3.6 + 0) i + (−1.6 + 2.9 − 3.7) j
R = 0.6i − 2.4j
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Homework Set 5 - Due Mon. Sept. 20
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Read Sections 1.8-1.10
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Do Problems 1.35, 1.44, 1.52 & 1.53
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