Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) Show that the distance between the points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) is \(\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)}\) (b) Simplify the Distance Formula for \(\theta_{1}=\theta_{2} .\) Is the simplification what you expected? Explain. (c) Simplify the Distance Formula for \(\theta_{1}-\theta_{2}=90^{\circ}\) Is the simplification what you expected? Explain.

Step-by-Step Solution

Answer

Yes, the given distance formula in polar co-ordinates is right, it simplifies to the absolute difference of the radial co-ordinates when the points are along the same line from origin and the magnitude of the sum of the vectors when they are perpendicular to each other.

Step 1: Validation of the Distance Formula

Consider two points A and B with polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2)\) respectively. Translate these polar coordinates to Cartesian coordinates. Point A becomes \((x_1,y_1)\) where \(x_1 = r_1 \cos \theta_1\) and \(y_1 = r_1 \sin \theta_1\). Similarly, Point B becomes \((x_2,y_2)\) where \(x_2 = r_2 \cos \theta_2\) and \(y_2 = r_2 \sin \theta_2\). The distance \(D\) between the two points A and B in Cartesian coordinates is given by the Euclidean distance formula, \(D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). Substituting the expressions for \(x_1\), \(y_1\), \(x_2\), \(y_2\) from the polar coordinates into this formula and simplifying, the result is: \(D = \sqrt{r_1^{2}+r_2^{2}-2 r_1 r_2 \cos (\theta_1-\theta_2)}\), which matches with the given formula.

Step 2: Simplifying the Distance Formula When the Angles are Same

When \(\theta_1 = \theta_2\), \(cos(\theta_1 - \theta_2) = cos(0) = 1\). Substituting this value in the distance formula: \(D = \sqrt{r_1^{2}+r_2^{2}-2 r_1 r_2} = \sqrt{(r_1 - r_2)^2}\), which is the absolute difference between the distances of the two points from the origin. This is as expected, as when the points lie in the same direction from the origin, the distance between them is simply the difference in their distances from the origin.

Step 3: Simplifying the Distance Formula When the Difference in Angles is 90 Degrees

When \(\theta_1 - \theta_2 = 90\) degrees, \(cos(\theta_1 - \theta_2) = 0\). Substitute this value in the distance formula: \(D = \sqrt{r_1^{2}+r_2^{2}}\), which is the magnitude of the sum of the vectors in a direction making a right angle with the vectors. This is as expected, considering the geometric interpretation of the points in polar coordinates.