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Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov

Lecture 4

Chapter 3

Introduction to vectors

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Physics I

Chapter 3:

Vectors and Scalars: Section 3.1 Addition/ Subtraction of Vectors/ Multiplication of a

Vector by a Scalar : Section 3.2 Vector Components: Section 3.3 Unit Vectors: Section 3.4

Today we are going to discuss:

Vector and Scalar

e.g. distance, speed, temperature, mass, time, density, volume

r, v , a

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

has only magnitude (no need in direction)

Vector quantity Scalar quantity

has both direction and magnitude

V=10 L

Addition of Vectors (1D)

If the vectors are in opposite directions

If the vectors are in the same direction

For vectors in one dimension, simple addition and subtraction are all that is needed. Easy!!!!

Addition of Vectors (2D). Graphical MethodsTriangle method.

“Tail-to-Tip” method Draw first vector Draw second vector, placing its tail at the tip of the first vector Resultant: Arrow from the tail of 1st vector to the tip of 2nd vector

The situation is somewhat more complicated in a case of two dimensions.

Addition of Vectors (2D). Graphical MethodsParallelogram method.

The two vectors, , are drawn as the sides of the parallelogram The resultant, , is its diagonal

Commutative property of vectors

Subtraction of vectors

is a vector with the same magnitude as but in the oppositedirection. So we can rewrite subtraction as additionB

So, we add the negative vector.

Multiplication of a Vector by a Scalar

A vector can be multiplied by a scalar b(positive); the result is a vector that has the same direction but a magnitude .

If b is negative, the resultant vector points in the opposite direction.

Addition of three or more vectorsCan use “tip to tail” for more than 2 vectors

Order of addition does not matter

Vector componentsThe graphical addition of vectors is not an especially good way to find

quantitative results.

Here we will need a coordinate system to resolve a vector into components along

mutually perpendicular directions.

Determining vector components (Vx, Vy)If we know magnitude, V and direction, θ

Assume that θ is measured counterclockwise from the positive x-axis In 2D, we can always write any vector as the sum of a vector in the x-

direction, and one in the y-direction.V

cos VVx V

sinVVy

Given Vx and Vy , we can find V, θ Vx, Vy are the legs of the right triangle Vector as the hypotenuse. So, the magnitudes of the vectors satisfy the Pythagorean Theorem.

222yx VVV

22yx VVV

VV1tan so

Determining the direction, θ of a vectorWe can describe vector’s direction by its angle relative to some reference direction.

Assume that θ is measured from the positive x-axis Θ is positive if measured counterclockwise from +x

I IIIII IV

Alternative ways of determining θWe can describe vector’s direction by its angle relative

to some other reference direction.

θ is measured clockwise from the positive x‐axis (Θ is negative)

θ is measured clockwise from the negative x‐axis

(you can say: ‐30 deg or 30 deg below +x axis)

(you can say: 30 deg above –x axis)

A vector is given by its vector components:

Write the vector in terms of magnitude and direction.4,2 yx VV

47.42042 22 V

yx VVV

tan 224tan

magnitude

axisxabove 11763180180

632tan 1

axisxfrom

Example A hiker on a trail

Adding vectors by components

Given and , how can we find ?

21 VVV

= yx VVV ,

yx VVV 111 ,

yx VVV 222 ,

yyxx VVVV 2121 ,

Adding corresponding components

Unit Vectors

As we said before, a vector has both magnitude and direction. Now, it’s time to simplify a notation of direction: Let’s introduce unit vectors

vectorsunitasknownkji ˆ,ˆ,ˆ

• They point along major axes of our coordinate system

= = =1 • Unit vector has a magnitude of 1

Writing a vector with unit vectorsWriting a vector with unit vectors is equivalent to

multiplying each unit vector by a scalar

• If a vector has components:

• In unit vector notation, we write

3,4 yx VV

jiV ˆ3ˆ4

yx VVV

),( yx VV

)3,4(iVxˆ

jViV yxˆˆ

D1 (2500m)i (500m) jD2 (500m)i (700m) j (700m)kD3 (600m) jD4 (500m)k

D (3000m)i (1800m) j (200m)k

(2500m)i (500m) j

(500m)i (700m) j (700m)k

(600m) j

(500m)k

What is the hiker’s total displacement?

The first leg is a flat hike to the foot of the mountain: ----------------------------On the second leg, he climbs the mountain:----------------------------------

On the third, he walks along a plateau: ---

Then, he falls off a cliff: ---------------------

Example A hiker on a trail

Thank you

Example 1

A vector is given by its magnitude and direction (V,) What is the x, y-component

of the vector?

mV 10axisxabove 30

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