Vectors in the Polar Plane Help
Vectors in the Polar Plane
In the mathematician’s polar coordinate plane, vectors a and b can be denoted as rays from the origin (0,0) to points ( θ a , r a ) and ( θ b , r b ) as shown in Fig. 6-7.
Magnitude, Direction, and Sum
Magnitude And Direction
The magnitude and direction of vector a in the polar coordinate plane are defined directly:
By convention, the following restrictions hold:
The sum of two vectors a and b in polar coordinates is best found by converting them into their equivalents in rectangular ( xy -plane) coordinates, adding the vectors according to the formula for the xy -plane, and then converting the resultant back to polar coordinates. To convert vector a from polar to rectangular coordinates, these formulas apply:
x a = r a cos θ a
y a = r a sin θ a
To convert vector a from rectangular coordinates to polar coordinates, these formulas apply:
Multiplication By Scalar and Dot Product
Multiplication By Scalar
In two-dimensional polar coordinates, let vector a be defined by the coordinates ( θ,r ) as shown in Fig. 6-8. Suppose a is multiplied by a positive real scalar k. Then the following equation holds:
k a = (θ,kr)
If a is multiplied by a negative real scalar –k, then:
– k a = [(θ+ 180°), kr ]
for θ in degrees. For θ in radians, the formula is:
–k a = [(θ + π),kr]
The addition of 180° (π rad) to θ reverses the direction of a . The same effect can be produced by subtracting 180° (π rad) from θ .
Let r a be the radius of vector a , and r b be the radius of vector b in the polar plane. Then the dot product of a and b is given by:
a · b = |a||b| cos ( θ b – θ a )
= r a r b cos ( θ b – θ a )
Vectors in the Polar Plane Practice Problems
Consider the vector a c = ( x a , y a ) = (3,4) in Cartesian coordinates. What is the equivalent vector a p = ( θ a , r a ) in mathematician’s polar coordinates? Express values to the nearest hundredth of a linear unit, and to the nearest degree.
Use the conversion formulas above. First find the direction angle θ a . Because x a > 0, we use this formula:
θ a = arctan ( y a / x a )
= arctan (4/3)
= arctan 1.333
Solving for r a , we proceed as follows:
Therefore, a p = ( θ a , r a ) = (53°,5.00).
Consider the vector b p = ( θ b , r b ) = (200°,4.55) in mathematician’s polar coordinates. Convert this to an equivalent vector b c = ( x b , y b ) in Cartesian coordinates. Express your answer to the nearest tenth of a unit for both coordinates x b and y b .
Use the conversion formulas above. First, solve for x b :
x b = r b cos θ b
= 4.55 cos 200°
= 4.55 × (–0.9397)
Next, solve for y b :
y b = r b sin θ b
= 4.55 sin 200°
= 4.55 × (–0.3420)
Therefore, b c = ( x b , y b ) = (–4.3, –1.6).
Practice problems for these concepts can be found at: Three-Space and Vectors Practice Test