Vectors

• Vectors and Scalars

• Properties of Vectors

• Components of a Vector and Unit Vectors

• Homework

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Vectors and Scalars

• Vector - quantity that has magnitude and direction

– e.g. displacement, velocity, acceleration, force

• Scalar - quantity that has only magnitude

– e.g. Time, mass, energy

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Displacement Vector

As a particle moves from A to B along the path repre-sented by the dashed curve, its displacement is the vectorshown 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 thevector that runs from the tail of A to the head of B.

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Commutative Property of Vector Addition

• The vectorR resulting from the addition of the vectorsA and B is the diagonal of a parallelogram of sides Aand B.

• Vector addition is commutative, that is A + B = B + A.

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Associative Property of Vector AdditionA+(B+C) = (A+B)+C

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Subtraction of Vectors

• To subtract vector B from vector A, simply add thevector -B to vector A.

• The vector -B is equal in magnitude and opposite indirection to the vector B.

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Components of a Vector

A vector A lying in the xy plane can be represented by itscomponent vectors Ax and Ay.

Ax = A cos θ Ay = A sin θ

A =√

A2x + A2

y tan θ =Ay

Ax

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Unit Vectors

• 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 per-pendicular vectors and the magnitude of each unitvector is one

– |i|=|j|=|k|=1

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Vectors in Component Form

A vector A lying in the xy plane has component vectorsAxi and Ayj where Ax and Ay are the components of A.

A=Axi + Ayj

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Example 1

A small plane leaves an airport on an overcast day andlater is sighted 215 km away, in a direction making anangle of 22◦ east of north. (a) How far east and northis the airplane from the airport when sighted? (b) Usinga coordinate system with the y-axis pointing north andthe x-axis east, write the position of the airplane in unitvector notation.

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Example 1 Solution

A small plane leaves an airport on an overcast day andlater is sighted 215 km away, in a direction making anangle of 22◦ east of north. (a) How far east and north isthe airplane from the airport when sighted?

rx

ry

y

x

r

θ

N

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 pointingnorth and the x-axis east, write the position of the air-plane in unit vector notation.

r = rxi + ryj = (81 km) i + (199 km) j

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Vector Addition Using Components

R = A + B

Rxi + Ryj = (Axi + Ayj) + (Bxi + Byj)

Rxi + Ryj = (Ax + Bx) i + (Ay + By) j

Rx = Ax + Bx Ry = Ay + By

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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 = Rxi + Ryj

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

• Read Sections 1.8-1.10

• Do Problems 1.35, 1.44, 1.52 & 1.53

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