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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).

     - the initial point

             - the terminal point

     - the magnitude of the vector (length)

    Zero vector ( )

    • 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:

    Parallel vectors

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

    Translation

    • A vector  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.

    * 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 :

    The product of a scalar  and a vector  is a vector  whose magnitude is  times the magnitude of the original vector  and depending on the sign of , the direction is:

    • the same as  , if
    • opposite of  , if

    If either  or , then

    Sum of vectors

    Vector addition  

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

    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   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.

    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:

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

    Difference.png

    Note: the vector difference   is equivalent to the vector sum   . This means we can add the vector  with the opposite of the vector .

    Difference2.png


    1. (Example 2) Consider  the vectors  and . Sketch each of the stated  vectors.

    1.  (using the Triangle Rule)

    1.  (using the Parallelogram Rule)

    1.  


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

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

    Graph.png

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

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

    Graph.png

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

    1. (Example 4) Express vector  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    is a radius vector in 2D space with initial point at the origin and terminal point at ,  then its component form  is:

     - component in the direction of the -axis

     - component in the direction of the -axis

    Component form of an arbitrary vector

    • If    is a vector with initial point  and terminal point at ,  then its component form  is:

    Magnitude of a vector

    The magnitude of a standard-position vector  is

    The magnitude of  is

    Operations on Vectors in Component Form

    Let  and  be vectors, and let  be a scalar.

    • Scalar multiplication :
    • Vector addition :

    Properties of Vector Operations

    1. Commutative Property

    1. Associative Property

    1. Additive identity property

    1. Additive inverse property

    1. Associativity of scalar multiplication

    1. Distributive property

    1. Distributive property

    1. Identity and zero properties

     ,  


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

    Using algebra, find

    Finding the components

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

     - angle between the vector and the positive -axis

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

    Unit vector

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

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

    Proof :  For any scalar , we have . In this case , so . Therefore, .

    1. (Example 7) Let .
    1. Find a unit vector with the same direction as .
    2. Find a vector  with the same direction as  such that .

    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    and .

    Thus,

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

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


    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 . 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 -axis.


    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?

    Reference :


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

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