Unit 2 - Objective 1 - Intro to Vectors
Quick Review

A vector is a quantity with both magnitude and direction. Vectors are used to represent quantities such as force, torque, displacement, and electrical current. A scalar is a representation of quantity that just requires a number like length, mass, volume, and electrical charge.

Vectors can be represented graphically with an arrow. For the vector A below, the point O is the initial (starting) point or tail. The point P is the terminal (ending) point or head of the vector A.



The length of the arrow, drawn to scale, is the magnitude of the vector, denoted with vertical bars, |A|. The arrow's orientation in the plane shows its direction. Vectors with the same magnitude and same direction (parallel) are equivalent vectors. All 4 vectors above represent the same vector A.

We move the vector in the coordinate plane so that its initial point is at the origin, (0,0), then the terminal point will correspond to some point (a,b) in the plane. This is called the standard position of the vector. Here a is the vector's x-component, and b is the y-component. The notation for a vector in component form is <a,b> or (a,b). Vector A above is <4,2> or (4,2).

Graphical representation of a scalar times a vector:

Given a vector B , then:

  • 3B is a vector in the same direction, 3 times as long

  • -2B is a vector twice as long as B, in the opposite direction

  • -B = (-1)B is the same length as B, in the opposite direction

  • 0B = a Zero vector = <0,0> = a point

Graphical representation of the sum or resultant of two vectors:

A + B is the resultant that would give the same result as moving along A, followed by moving along B. It can be represented graphically by one of two methods.

  1. Move B so that its initial point corresponds to A's terminal point. The resultant, A + B, begins at the initial point of A and ends at the terminal point of B.

    This method can be used for adding more than 2 vectors.

  2. An alternate way for representing A + B is by moving A and B so that their initial points (tails) coincide. Draw a parallelogram. The resultant vector is the vector with the same common tail, the diagonal of the parallelogram. This should look like the same vector sketched by method #1 above.

Examples


Graphically represent the following:
1) A + B

2) -2C

3) A - C = A + (-C)


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