Questions: Find the distance (d(P1, P2)) between the points (P1) and (P2). (P1=(0.1,-0.5)) (P2=(3.5,3.5))

Transcript text: Find the distance $d\left(P_{1}, P_{2}\right)$ between the points $P_{1}$ and $P_{2}$. \[ \begin{array}{l} P_{1}=(0.1,-0.5) \\ P_{2}=(3.5,3.5) \end{array} \]

Solution

Solution Steps

To find the distance between two points $ P_1 $ and $ P_2 $ in a 2D plane, we can use the Euclidean distance formula: \[ d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where $ P_1 = (x_1, y_1) $ and $ P_2 = (x_2, y_2) $.

Step 1: Identify the Coordinates

Given the points: \[ P_1 = (0.1, -0.5) \] \[ P_2 = (3.5, 3.5) \]

Step 2: Apply the Euclidean Distance Formula

The Euclidean distance formula between two points $ P_1 = (x_1, y_1) $ and $ P_2 = (x_2, y_2) $ is: \[ d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Substitute the Coordinates

Substitute $ x_1 = 0.1 $, $ y_1 = -0.5 $, $ x_2 = 3.5 $, and $ y_2 = 3.5 $ into the formula: \[ d(P_1, P_2) = \sqrt{(3.5 - 0.1)^2 + (3.5 + 0.5)^2} \]

Step 4: Simplify the Expression

Calculate the differences: \[ d(P_1, P_2) = \sqrt{(3.4)^2 + (4.0)^2} \]

Step 5: Compute the Squares

\[ d(P_1, P_2) = \sqrt{11.56 + 16} \]

Step 6: Sum the Squares

\[ d(P_1, P_2) = \sqrt{27.56} \]

Step 7: Calculate the Square Root

\[ d(P_1, P_2) \approx 5.250 \]