**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:

**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:

**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* .

**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* _{b} – *θ* _{a} )

= *r* _{a} *r* _{b} cos ( *θ* _{b} – *θ* _{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:

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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