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Untitled Page 01

  • Page ID
    30083
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    2.1 Vectors in Plane

    Vector

    • Vector: a quantity that has both a magnitude  and direction *
    • We represent vectors as directed line segments.  

    Notation : draw an arrow above the name of the vector (or write in bold).

    image69.png

    image72.png  - the initial point

            image71.png  - the terminal point

    image76.png

    image74.png  - the magnitude of the vector image80.png (length)

    Zero vector ( image78.png )

    • A vector whose initial and terminal point is the same point.
    • The zero vector is the only vector without direction (i.e. it has any arbitrary direction)
    • The length of the zero vector is zero: image85.png

    Parallel vectors

    • Two nonzero vectors image44.png  and image89.png are parallel if the lines going through them are parallel.
    • Parallel vectors point either in the same or in the opposite direction.

    image86.png

    image88.png

    Translation

    • A vector image7.png  is translated when it is moved parallel to itself without changing its length or direction. Translated vectors have the same length and direction but different endpoints.

    Equivalent vectors

    • Two vectors are equivalent if they have the same length and direction .
    • If two vectors are equivalent, then either of these vectors can be obtained by translating the other vector.

    image90.png

    * Other common quantities that we encounter in mathematics and physics are scalars, which are simply numbers (so quantities that have only a value, a magnitude, but no direction).


    Operations Involving Vectors

    Multiplication of a vector by a scalar

    Scalar multiplication : image91.png

    The product of a scalar image35.png  and a vector image1.png  is a vector  whose magnitude is image92.png  times the magnitude of the original vector image1.png  and depending on the sign of image35.png , the direction is:

    • the same as image1.png  , if image63.png
    • opposite of image1.png  , if image65.png

    If either image66.png  or image67.png , then image68.png

    Sum of vectors

    Vector addition   image16.png

    Triangle Method: place the initial point of the second vector ( image5.png ) at the terminal point of the first vector ( image1.png ). The resultant vector representing the sum image16.png   of the two vectors is then the vector whose initial point coincides with the initial point of image1.png  and the terminal point coincides with the terminal point of image5.png .

    TriangleMethod.PNG

    Parallelogram Method: place the two vectors so that they have the same initial point and then draw a parallelogram in which the two vectors form two adjacent sides. The sum image16.png   of the two vectors is then the diagonal of the parallelogram as shown in the picture.

    ParallelogramMethod.PNG

    Triangle Inequality

    In a triangle, the length of any one side is less than the sum of the lengths of the remaining sides.

    image20.png

    Note : If the two vectors are parallel and point in the same direction, then the magnitude of the resultant vector equals the sum of the magnitudes of the two component vectors.

    Difference of vectors

    Vector Subtraction: image3.png

    Triangle Method: The difference image3.png   of two vectors is the vector from terminal point of image5.png  to the terminal point of image1.png .

    Difference.png

    Note: the vector difference image3.png   is equivalent to the vector sum   image4.png . This means we can add the vector image1.png  with the opposite of the vector image5.png .

    Difference2.png


    1. (Example 2) Consider  the vectors image7.png  and image8.png . Sketch each of the stated  vectors. image46.png

    1. image26.png

    1. image27.png  (using the Triangle Rule)

    1. image27.png  (using the Parallelogram Rule)

    1. image28.png  


    1. (Example 3) Are image7.png  and image8.png  equivalent vectors?
    1. image7.png  has initial point (3, 2) and terminal point (7, 2).

    image8.png  has initial point (1, -4) and terminal point (1, 0).

    Graph.png

    1. image7.png  has initial point (0, 0) and terminal point (1, 1).

    image8.png  has initial point (-2, 2) and terminal point (-1, 3).

    Graph.png

    1. (Exercise. 3) Which of the following vectors are equivalent?

    image23.png

    1. (Example 4) Express vector image7.png  with initial point (-3, 4) and terminal point (1, 2) in component form.

    Graph.png

    Component Form of a Vector

    Standard-position vector (radius vector).

    • A vector whose initial point coincides with the origin .
    • If   image1.png  is a radius vector in 2D space with initial point at the origin and terminal point at image24.png ,  then its component form  is:

    image25.png

    image9.png  - component in the direction of the image9.png -axis

    image21.png  - component in the direction of the image21.png -axis

    Component form of an arbitrary vector

    • If   image1.png  is a vector with initial point image50.png  and terminal point at image51.png ,  then its component form  is:

    image52.png

    Magnitude of a vector

    The magnitude of a standard-position vector image25.png  is

    image53.png

    image54.png

    The magnitude of image52.png  is

    image47.png

    Operations on Vectors in Component Form

    Let image48.png  and image49.png  be vectors, and let image35.png  be a scalar.

    • Scalar multiplication : image37.png
    • Vector addition : image39.png

    Properties of Vector Operations

    1. Commutative Property

    image40.png

    1. Associative Property

    image42.png

    1. Additive identity property

    image43.png

    1. Additive inverse property

    image45.png

    1. Associativity of scalar multiplication

    image29.png

    1. Distributive property

    image31.png

    1. Distributive property

    image32.png

    1. Identity and zero properties

    image33.png  ,   image79.png


    1. (Example 5) Let image7.png  be the vector with initial point (2, 5) and terminal point (8, 13) and let image83.png .
    1. Express image7.png  in component form and find image87.png .

    Using algebra, find

    1. image27.png

    1. image14.png

    1. image13.png

    Finding the components

    Given the magnitude and direction of a vector, we can find the Cartesian components of a vector:

    image17.png image81.png

    image15.png

    image11.png  - angle between the vector and the positive image9.png -axis

    image12.png

    1. (Example 6) Find the component form of a vector with magnitude 4 that forms an angle of image10.png  with the image9.png -axis. image55.png

    Unit vector image64.png

    Unit vector  – a vector of unit length (magnitude of 1) in some given direction.

    For a nonzero vector image1.png , a unit vector in the same direction is obtained by scalar multiplication by the reciprocal of magnitude [1] :

    image61.png

    image62.png

    Proof :  For any scalar image35.png , we have image58.png . In this case image57.png , so image60.png . Therefore, image59.png .

    1. (Example 7) Let image56.png .
    1. Find a unit vector with the same direction as image7.png .
    2. Find a vector image8.png  with the same direction as image7.png  such that image77.png .

    Standard unit vectors

    If we use the Cartesian (rectangular) coordinate system, the two components of a vector represent the horizontal  and the vertical  component of a vector, so it is convenient to use the standard unit vectors   image84.png  and image82.png .

    image73.png

    Thus,

    image70.png

    1. (Example 8)
    1. Express the vector image75.png  in terms of standard unit vectors.

    1. Vector image44.png is a unit vector that forms an angle of image41.png  with the positive image9.png -axis. Express image44.png  in terms of the standard unit vectors.

    image36.png


    1. (Example 9) Jane’s car is stuck in the mud. Lisa and Jed come along in a truck to help pull her out. They attach one end of a tow strap to the front of the car and the other end to the truck’s trailer hitch, and the truck starts to pull. Meanwhile, Jane and Jed get behind the car and push. The truck generates a horizontal force of 300 lb on the car. These forces can be represented as vectors. The angle between these vectors is image34.png . Find the resultant force (the vector sum) and give its magnitude to the nearest tenth of a pound and its direction angle with respect to the positive image9.png -axis.

    image38.png


    1. (Example 10) An airplane flies dues west at an airspeed of 425 mph. The wind is blowing from the northeast at 40 mph. What is the ground speed of the airplane? What is the bearing of the airplane?

    image30.png

    Reference :


    [1]  The process of finding a unit vector in a given direction is called normalization .


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