Introduction to Vectors in the Cartesian Plane
A vector is a mathematical expression for a quantity with two independent properties: magnitude and direction . The direction, also called orientation , is defined in the sense of a ray, so it “points” somewhere. Vectors are used to represent physical variables such as distance, velocity, and acceleration. Conventionally, vectors are denoted by boldface letters of the alphabet. In the xy plane, vectors a and b can be illustrated as rays from the origin (0,0) to points ( x _{a} , y _{a} ) and ( x _{b} , y _{b} ) as shown in Fig. 92.
Equivalent Vectors
Occasionally, a vector is expressed in a form that begins at a point other than the origin (0,0). In order for the following formulas to hold, such a vector must be reduced to socalled standard form , where it begins at the origin. This can be accomplished by subtracting the coordinates ( x _{0} , y _{0} ) of the starting point from the coordinates of the end point ( x _{1} , y _{1} ). For example, if a vector a * starts at (3, −2) and ends at (1, −3), it reduces to an equivalent vector a in standard form:
a = {(1 − 3), [−3 −(−2)]}
= (−2, −1)
Any vector a * that is parallel to a , and that has the same length and the same direction (or orientation) as a , is equal to vector a . A vector is defined solely on the basis of its magnitude and its direction (or orientation). Neither of these two properties depends on the location of the end point.
Defining the Magnitude and Direction of a Vector
Magnitude
The magnitude (also called the length, intensity , or absolute value ) of vector a , written  a  or a , can be found in the Cartesian plane by using a distance formula resembling the Pythagorean theorem:
Direction
The direction of vector a , written dir a , is the angle θ _{a} that vector a subtends as expressed counterclockwise from the positive x axis:
dir a = θ _{a}
The tangent of the angle θ _{a} is equal to y _{a} / x _{a} . Therefore, θ _{a} is equal to the inverse tangent , also called the arctangent (abbreviated arctan or tan ^{−1} ) of y _{a} / x _{a} . Therefore:
dir a = θ _{a} = arctan ( y _{a} / x _{a} ) = tan ^{−1} ( y _{a} / x _{a} )
By convention, the angle θ _{a} is reduced to a value that is at least zero, but less than one full counterclockwise revolution. That is, 0° ≤ θ _{a} < 360° (if the angle is expressed in degrees), or 0 rad ≤ θ _{a} < 2π rad (if the angle is expressed in radians).
Sum
The sum of two vectors a and b , where a = ( x _{a} , y _{a} ) and b = ( x _{b} , y _{b} ), is given by the following formula:
a + b = [( x _{a} + x _{b} ), ( y _{a} + y _{b} )]
This sum can be found geometrically by constructing a parallelogram with a and b as adjacent sides. Then a + b is the diagonal of this parallelogram (Fig. 92).
Fig. 92
Two vectors in the Cartesian plane. They are added using the “parallelogram method.”

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