Questions: 1. Charles is exploring the Stellarium app and found SELF-PACED PRACTICE PROBLEM 1 vector addition the moon 30 units North of Vega while Venus is 70 units East of the moon. How far is Vega from Venus. Indicate both the magnitude and the direction

Solution Steps

We need to determine the distance and direction from Vega to Venus. The problem provides the following information:

• The moon is 30 units North of Vega.
• Venus is 70 units East of the moon.

Let's set up a coordinate system where:

• Vega is at the origin \((0, 0)\).
• The moon is 30 units North of Vega, so its coordinates are \((0, 30)\).
• Venus is 70 units East of the moon, so its coordinates are \((70, 30)\).

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of Vega \((0, 0)\) and Venus \((70, 30)\): \[ d = \sqrt{(70 - 0)^2 + (30 - 0)^2} = \sqrt{70^2 + 30^2} = \sqrt{4900 + 900} = \sqrt{5800} \] \[ d \approx 76.1577 \]

The direction from Vega to Venus can be found using the angle \(\theta\) with respect to the positive x-axis (East direction). The angle \(\theta\) is given by: \[ \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{30}{70} \] \[ \theta = \tan^{-1}\left(\frac{30}{70}\right) \approx 23.1986^\circ \]

Final Answer