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    A Has As Well As - Some Vector Quantities

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    Mykee Suizo
    1. Vectors can be added graphically by placing the tail of one vector at the head of another. They can also be added by resolving into x and y components and adding the components. 2. To find the magnitude and direction of the resultant vector, use the Pythagorean theorem and trigonometric functions like tangent. 3. Vectors can be multiplied by a scalar, which scales the magnitude but keeps the direction the same. Negative vectors point in the opposite direction.

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    A Has As Well As - Some Vector Quantities

    Uploaded by

    Mykee Suizo
    57 views42 pages
    1. Vectors can be added graphically by placing the tail of one vector at the head of another. They can also be added by resolving into x and y components and adding the components. 2. To find the magnitude and direction of the resultant vector, use the Pythagorean theorem and trigonometric functions like tangent. 3. Vectors can be multiplied by a scalar, which scales the magnitude but keeps the direction the same. Negative vectors point in the opposite direction.

    Original Description:

    About Vectors

    Original Title

    Vector

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    © Attribution Non-Commercial (BY-NC)

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    1. Vectors can be added graphically by placing the tail of one vector at the head of another. They can also be added by resolving into x and y components and adding the components. 2. To find the magnitude and direction of the resultant vector, use the Pythagorean theorem and trigonometric functions like tangent. 3. Vectors can be multiplied by a scalar, which scales the magnitude but keeps the direction the same. Negative vectors point in the opposite direction.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    57 views42 pages

    A Has As Well As - Some Vector Quantities

    Uploaded by

    Mykee Suizo
    1. Vectors can be added graphically by placing the tail of one vector at the head of another. They can also be added by resolving into x and y components and adding the components. 2. To find the magnitude and direction of the resultant vector, use the Pythagorean theorem and trigonometric functions like tangent. 3. Vectors can be multiplied by a scalar, which scales the magnitude but keeps the direction the same. Negative vectors point in the opposite direction.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 42

    Vectors and Scalars

    A vector has magnitude as well as direction. Some vector quantities: displacement, velocity, force, momentum A scalar has only a magnitude. Some scalar quantities: mass, time, temperature

    Distance: A Scalar Quantity


    Distance is the length of the actual path taken by an object.
    s = 20 m
    A

    A scalar quantity:

    Contains magnitude only and consists of a number and a unit. (20 m, 40 mi/h, 10 gal)

    DisplacementA Vector Quantity


    Displacement is the straight-line separation of two points in a specified direction.
    D = 12 m, 20o
    A
    q

    A vector quantity:

    Contains magnitude AND direction, a number, unit & angle. (12 m, 300; 8 km/h, N)

    More about Vectors


    A vector is represented on paper by an arrow 1. the length represents magnitude 2. the arrow faces the direction of motion

    3-2 Addition of VectorsGraphical Methods

    For vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates.

    Applications of Vectors
    VECTOR ADDITION If 2 similar vectors point in the SAME direction, add them. Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started?

    54.5 m, E +

    30 m, E

    84.5 m, E

    Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION.

    Applications of Vectors
    VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT. Example: A man walks 54.5 meters east, then walks 30 meters west. Calculate his displacement relative to where he started? 54.5 m, E 30 m, W +

    24.5 m, E

    Easy Adding

    All these planes have the same reading on their speedometer. (plane speed not speed with respect to the ground (actual speed)

    What factor is affecting their velocity?

    Addition of VectorsGraphical Methods


    If the motion is in two dimensions, the situation is somewhat more complicated.
    Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem.

    Perpendicular Vectors
    When 2 vectors are perpendicular, you may use the Pythagorean theorem.
    A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT. 55 km, N
    c2 a 2 b2 c a 2 b2 c Resultant 952 552

    95 km,E

    c 12050 109.8 km

    Example
    A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement.

    - = 12 m, W 6 m, S 20 m, N 35 m, E R
    q

    23 m, E

    - =

    14 m, N
    R 14 2 232 26.93m 14 .6087 23 q Tan1 (0.6087) 31.3 Tanq

    14 m, N

    26.93m31.3 26.93m,31.3

    23 m, E
    The Final Answer:

    Addition of VectorsGraphical Methods


    Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.

    To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words). Terminal point of w

    vw
    v

    Initial point of v

    Move w over keeping the magnitude and direction the same.

    A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

    3v

    v v

    v
    The negative of a vector is just a vector going the opposite way.

    v v

    v u w

    Using the vectors shown, find the following:

    uv

    3w
    w w

    u v

    2u 3w v v
    u u w w w

    uv
    u v

    3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar


    In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction.

    Then we add the negative vector.

    3-4 Adding Vectors by Components


    Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other.

    3-4 Adding Vectors by Components


    Remember: soh cah toa

    If the components are perpendicular, they can be found using trigonometric functions.

    Adding Vectors by Components


    The components are effectively one-dimensional, so they can be added arithmetically.

    3-4 Adding Vectors by Components


    Adding vectors:
    1. Draw a diagram; add the vectors graphically. 2. Choose x and y axes. 3. Resolve each vector into x and y components. 4. Calculate each component using sines and cosines.

    5. Add the components in each direction.


    6. To find the length and direction of the vector, use: and

    3-4 Adding Vectors by Components


    Example 3-2: Mail carriers displacement. A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0 south of east for 47.0 km. What is her displacement from the post office?

    3-4 Adding Vectors by Components


    Example 3-3: Three short trips. An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is 45 south of east (315) for 440 km; and the third leg is at 53 south of west (233), for 550 km, as shown. What is the planes total displacement?

    Analytical Method
    Polar Form of Vectors
    = ,
    =

    Example

    =
    = 10 , 450 = 10 = 450

    Caution
    Addition of vectors in polar form cannot be done algebraically Ex. A = 5 km, 45 deg B = 4 km, 135 deg C = 3 km, 270 deg R = 12 km, 450 deg
    Vectors can only be added algebraically if they are parallel or antiparallel

    Component Form
    = ,
    = cos + sin
    and

    = 10 , 450
    = 10 (45) + 10 sin 45 = 7.07 + 7.07

    = 10 , 450 = 10 (45) + 10 sin 45 = 7.07 + 7.07

    10

    7.07

    7.07

    = 20 , 1200 = 20 (120) + 20 sin 120 = 10 + 17.32

    20 17.32

    10

    = 7.07 + 7.07 = 10 + 17.32

    R = -2.93 km x + 24.39 km y

    17.32 10

    7.07
    7.07

    Graphical Representation of the Analytical Method


    = 12 km, 30 deg
    = 6 km, 60 deg

    + =

    = +

    = +

    1 +2 + 3 =

    Component Method
    x D1 D2 D3 620km, 0 deg 440km, 315 deg 550 km, 233 deg 620km (cos 0) 440km (cos 315) 550km (cos 233) x D1 D2 D3 620km, 0 deg 440km, 315 deg 550 km, 233 deg R 620km 311km -331 km 600 km 0km -311 km -439 km -750 km y 620km (sin 0) 440km (sin 315) 550km (sin 233) y

    Adding vectors:
    1. Draw a diagram; add the vectors graphically. 2. Choose x and y axes. 3. Resolve each vector into x and y components. 4. Calculate each component using sines and cosines. 5. Add the components in each direction. 6. To find the magnitude of the vector, use:

    2 + 2

    x +y

    Direction:
    x y

    Q2

    Q1

    +x +y

    Q3

    Q4

    1. Resolve what quadrant is the vector pointing at?

    +x y

    2. Get the Reference Angle = 3. if


    Quadrant 1 =

    Quadrant 2 =
    Quadrant 3 = + Quadrant 4 =

    From Component to Polar


    Magnitude of R = 960
    Angle of R =

    6002 + 750

    1 750 600

    =51 deg at Q4

    360-51

    = 309 degrees

    = 600 750
    600

    750

    Scaling Vectors
    31 22 + 43 =
    D1 D2 D3 620km, 0 deg 440km, 315 deg 550 km, 233 deg 3*D1 = 1860 km, 0 deg 2*D2 = 880 km, 315 deg +4*D3 = 2200 km, 233 deg

    x
    3*D1 = 1860 km, 0 deg 2*D2 = 880 km, 315 deg 4*D3 = 2200 km, 233 deg R

    Problem Set
    = 4 , 30 = 13 , 120 = 10 , 240 = 7 , 310 Find the magnitude and direction of the following 1. + + + = 2. + + = 3. + + = 4. + + = 5. + = 6. + =

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