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PHY1012F VECTORS. Gregor Leigh [email protected] VECTORS. Resolve vectors into components and reassemble components into a single vector with magnitude and direction. Make use of unit vectors for specifying direction.

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PHY1012F VECTORS

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Phy1012f vectors

PHY1012FVECTORS

Gregor [email protected]


Vectors

VECTORS

  • Resolve vectors into components and reassemble components into a single vector with magnitude and direction.

  • Make use of unit vectors for specifying direction.

  • Manipulate vectors (add, subtract, multiply by a scalar) both graphically (geometrically) and algebraically.

Learning outcomes:At the end of this chapter you should be able to…


Vectors1

VECTORS

  • Using only positive and negative signs to denote the direction of vector quantities is possible only when working in a single dimension (i.e. rectilinearly).

  • In order to deal with direction when describing motion in 2-d (and later, 3-d) we manipulate vectors using either graphical (geometrical) techniques, or the algebraic addition of vector components.


Scalars and vectors

SCALARS and VECTORS

  • Scalar

– A physical quantity with magnitude (size) but no associated direction.E.g. temperature, energy, mass.

Vector

– A physical quantity which has both magnitude AND direction.E.g. displacement, velocity, force.


Vector representation and notation

Algebraically, we shall distinguish a vector from a scalar by using an arrow over the letter, .

VECTOR REPRESENTATION and NOTATION

  • Graphically, a vector is represented by a ray.The length of the ray represents the magnitude, while the arrow indicates the direction.

The important information is in the direction and length of the ray – we can shift it around if we do not change these.

Note:r is a scalar quantity representing the magnitude of vector , and can never be negative.


Graphical vector addition

GRAPHICAL VECTOR ADDITION

  • A helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement.

N

20 km

60°

10°

15 km

 = 74°


Multiplying a vector by a scalar

MULTIPLYING A VECTOR BY A SCALAR

  • Multiplying a vector by a positive scalar gives another vector with a different magnitude but the same direction:

Notes:

  • B = cA. (c is the factor by which the magnitude ofis changed.)

  • lies in the same direction as .

  • (Distributive law).

  • If c is zero, the product is the directionless zero vector, or null vector.


Vector components

  • y

  • x

  • z

VECTOR COMPONENTS

  • Manipulating vectors geometrically is tedious.

  • Using a (rectangular) coordinate system, we can use components to manipulate vectors algebraically.

  • We shall use Cartesian coordinates, a right-handed system of axes:

(The (entire) system can be rotated – any which way – to suit the situation.)


Vector components1

VECTOR COMPONENTS

  • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)…


Vector components2

VECTOR COMPONENTS

  • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)…

“Running the movie backwards” resolves a single vector into two (or more!) components.


Vector components3

VECTOR COMPONENTS

  • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)…

“Running the movie backwards” resolves a single vector into two (or more!) components.


Vector components4

VECTOR COMPONENTS

  • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)…

“Running the movie backwards” resolves a single vector into two (or more!) components.

??!

Even if the number of components is restricted, there is still an infinite number of pairs into which a particular vector may be decomposed.

Unless…


Vector components5

VECTOR COMPONENTS

  • …by introducing axes, we specify the directions of the components.

  • y

is now constrained to resolve into and , at right angles to each other.

Note that, provided that we adhere to the right-handed Cartesian convention, the axes may be orientated in any way which suits a given situation.

  • x


Vector components6

  • y

  • x

VECTOR COMPONENTS

  • Resolution can also be seen as a projection of onto each of the axes to produce vector components and .

Ax, the scalar component of (or, as before, simply its component) along the x-axis …

  • has the same magnitude as .

  • is positive if it points right; negative if it points left.

  • remains unchanged by a translation of the axes (but is changed by a rotation).


Vector components7

  • y (m)

8

6

4

2

  • x (m)

0

0

2

4

6

8

VECTOR COMPONENTS

  • The components of are…

Ax = +6 m

Ay = +3 m


Vector components8

VECTOR COMPONENTS

  • y (m)

  • The components of are…

8

6

Ax = +6 m

Ay = +3 m

4

2

  • x (m)

-8

-6

-4

-2

0


Vector components9

VECTOR COMPONENTS

  • y (m)

  • The components of are…

-8

-6

-4

-2

  • x (m)

-2

Ax = +6 m

Ay = +3 m

-4

-6

-8


Vector components10

VECTOR COMPONENTS

  • y (m)

  • The components of are…

4

2

Ax = –2 m

Ay = +4 m

  • x (m)

-4

-2

2

-2


Vector components11

VECTOR COMPONENTS

  • y (m)

  • The components of are…

4

2

Ax = –6 m

Ay = –5 m

  • x (m)

-8

-4

4

-2


Vector components12

VECTOR COMPONENTS

The components of are…

PHY1012F

  • y (m)

4

  • x (m)

2

Ax = –6 m

Ay = –5 m

–8 m

–3 m

4

-4

-8

-4

20


Vector components13

PHY1010W

VECTOR COMPONENTS

  • y (m)

  • The components of are…

  • x (m)

Ax = +Acos m

Ay = –Asin m


Vector components14

PHY1010W

VECTOR COMPONENTS

  • y (m)

  • The components of are…

  • x (m)

Ax = –Asin m

Ay = –Acos m

Note that we can (re)combine components into a single vector, i.e. (re)write it in polar notation, by calculating its magnitude and direction using Pythagoras and trigonometry:

(On this slide!)


Unit vectors

  • y

1

  • x

1

UNIT VECTORS

  • Components are most useful when used with unit vector notation.

  • A unit vector is a vector with a magnitude of exactly 1 pointing in a particular direction:

  • A unit vector is pure direction – it has no units!

  • Vector can now be resolved and written as:


Unit vectors1

UNIT VECTORS

  • y (m/s)

Given a 12 m/s velocity vector which makes an angle of 60° with the negative x-axis, write the vector in terms of components and unit vectors.

vy = +vsin60°

v = 12 m/s

  • 60°

vx = –(12 m/s)cos60° vx= –6.00 m/s

  • x (m/s)

vx = –vcos60°

vy = +(12 m/s)sin60° vy= +10.4 m/s

Hence:


Algebraic addition of vectors

ALGEBRAIC ADDITION OF VECTORS

  • Suppose

and

ThusDx= Ax + Bx + Cx andDy= Ay + By + Cy

In other words, we can add vectors by adding their components, axis by axis, to determine a single resultant component in each direction. These resultants can then be combined, or simply presented in unit vector notation.


Algebraic addition of vectors1

  • y

  • x

ALGEBRAIC ADDITION OF VECTORS

  • The process of vector addition by the addition of components can visualised as follows:

By

Dy = Ay + By

Ay

=

+

Ax

Bx

Dx = Ax + Bx


Algebraic addition of vectors2

  • y

  • x

ALGEBRAIC ADDITION OF VECTORS

  • While it is often quite acceptable to present as

Dy = Ay + By

its polar form is easily reconstituted from Dx and Dy

using and

Dx = Ax + Bx


Graphical vector addition1

GRAPHICAL VECTOR ADDITION

  • A helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement.

N

20 km

60°

10°

15 km

 = 74°


Algebraic addition of vectors3

ALGEBRAIC ADDITION OF VECTORS

  • A helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement.

  • y

N

20 km

+15 sin10°

+2.60

+15 cos10°

+14.77

10°

–18.79

–20 cos20°

–20 sin20°

–6.84

15 km

–16.2

+7.93

  • x

20°


Algebraic addition of vectors4

N

  • y

  • x

ALGEBRAIC ADDITION OF VECTORS

  • A spelunker is surveying a cave. He follows a passage 100 m straight east, then 50 m in a direction 30° west of north, then

150 m at 45° west of south. After a fourth unmeasured displacement he finds himself back where he started. Determine the magnitude and direction of his fourth displacement.

30°

50 m

100 m

150 m

45°


Algebraic addition of vectors5

ALGEBRAIC ADDITION OF VECTORS

  • y

N

Rx = 100 – 25 –106 + Dx = 0

 Dx = 31

30°

Ry = 0 +43.3 –106 + Dy = 0

 Dy = 62.7

50 m

  • x

100 m

150 m

45°

100

0

–25

43.3

–106

–106

63.7

69.9

?

?

?

31

62.7

?


Vectors2

VECTORS

  • Resolve vectors into components and reassemble components into a single vector with magnitude and direction.

  • Make use of unit vectors for specifying direction.

  • Manipulate vectors (add, subtract, multiply by a scalar) both graphically (geometrically) and algebraically.

Learning outcomes:At the end of this chapter you should be able to…


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