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Vectors in the Polar Plane Help

By — McGraw-Hill Professional
Updated on Oct 3, 2011

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.

Three-Space and Vectors Vectors in the Polar Plane

Fig. 6-7. Two vectors a and b in the polar plane. The angles are θ a and θ b . All angles are expressed in radians. The radii are r a and r b .

Magnitude, Direction, and Sum

Magnitude And Direction

The magnitude and direction of vector a in the polar coordinate plane are defined directly:

Three-Space and Vectors Vectors in the Polar Plane Magnitude And Direction

By convention, the following restrictions hold:

Three-Space and Vectors Vectors in the Polar Plane Magnitude And Direction

Sum

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:

Three-Space and Vectors Vectors in the Polar Plane Sum

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]

Three-Space and Vectors Vectors in the Polar Plane Multiplication By Scalar

Fig. 6-8. Multiplication of a polar-plane vector a by a positive real scalar k , and by a negative real scalar –k. All angles are expressed in radians.

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 .

Dot Product

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 bθ a )

         = r a r b cos ( θ bθ a )

Vectors in the Polar Plane Practice Problems

Practice 1

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.

Solution 1

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

    = 53°

Solving for r a , we proceed as follows:

Three-Space and Vectors Vectors in the Polar Plane Dot Product

Therefore, a p = ( θ a , r a ) = (53°,5.00).

Practice 2

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 .

Solution 2

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)

    = –4.3

Next, solve for y b :

y b = r b sin θ b

    = 4.55 sin 200°

    = 4.55 × (–0.3420)

    = –1.6

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

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