Problem 3 - Physics for Scientists and Engineers with Modern Physics

Question

If \(V_{x}=7.80\) units and \(V_{y}=-6.40\) units, determine the magnitude and direction of \(\overrightarrow{\mathbf{V}}\).

Step-by-Step Solution

Answer

The magnitude is approximately 10.09 units, and the direction is \(-39.19^\circ\) from the positive x-axis.

Step 1: Understanding the Components

You are given the components of the vector \(\overrightarrow{\mathbf{V}}\): \(V_{x} = 7.80\) units and \(V_{y} = -6.40\) units. These represent the horizontal and vertical components of the vector, respectively.

Step 2: Calculating the Magnitude

To find the magnitude of vector \(\overrightarrow{\mathbf{V}}\), use the Pythagorean theorem: \[|\overrightarrow{\mathbf{V}}| = \sqrt{V_{x}^2 + V_{y}^2}\] Substitute the given values: \[|\overrightarrow{\mathbf{V}}| = \sqrt{(7.80)^2 + (-6.40)^2} = \sqrt{60.84 + 40.96} = \sqrt{101.80} \approx 10.09\] Thus, the magnitude of \(\overrightarrow{\mathbf{V}}\) is approximately 10.09 units.

Step 3: Finding the Direction

The direction \(\theta\) of vector \(\overrightarrow{\mathbf{V}}\) is the angle it makes with the positive x-axis, calculated using the tangent function: \[\theta = \tan^{-1}\left(\frac{V_{y}}{V_{x}}\right)\]Substitute the given values: \[\theta = \tan^{-1}\left(\frac{-6.40}{7.80}\right) \approx \tan^{-1}(-0.8205) \approx -39.19^\circ\]Since the angle is negative, it indicates the vector is below the x-axis. Therefore, the direction of \(\overrightarrow{\mathbf{V}}\) is approximately \(-39.19^\circ\) from the positive x-axis.

Key Concepts

Magnitude of a Vector

The magnitude of a vector refers to its length or size. It is a fundamental concept in vector analysis and is used to determine how much a vector spans in the space it's defined. To calculate the magnitude of a vector
like \(\overrightarrow{\mathbf{V}}\)
with components \(V_x\) and \(V_y\), we use the Pythagorean Theorem. This is expressed through the formula:

• \(|\overrightarrow{\mathbf{V}}| = \sqrt{V_{x}^2 + V_{y}^2}\)

This formula helps us find the magnitude by treating the vector's components as the sides of a right-angled triangle. Using this method allows us to directly compute the vector's length. It's similar to finding the hypotenuse of a triangle formed between the vector and its component lines. For instance, if \(V_{x} = 7.80\) and \(V_{y} = -6.40\),
the magnitude is calculated as:
\[|\overrightarrow{\mathbf{V}}| = \sqrt{(7.80)^2 + (-6.40)^2} = \sqrt{101.80} \approx 10.09\] So, for each specific direction, the vector spans approximately 10.09 units in magnitude.

Direction of a Vector

The direction of a vector indicates the orientation it has in its space, specifically regarding which way it "points". This direction is usually calculated as the angle with respect to a reference line, often the positive x-axis. To determine a vector's direction, you can use trigonometric functions, primarily the tangent function. Here is how you calculate it:

• \(\theta = \tan^{-1}\left(\frac{V_{y}}{V_{x}}\right)\)

This formula calculates the angle \(\theta\), using the inverse tangent, \(\tan^{-1}\), of the ratio of the vector's vertical component \(V_y\) to its horizontal component \(V_x\).
This gives you the direction in degrees from the x-axis. For instance, with \(V_{x} = 7.80\) and \(V_{y} = -6.40\), this becomes:
\[\theta = \tan^{-1}\left(\frac{-6.40}{7.80}\right) \approx -39.19^\circ\] The negative angle here indicates that the vector is positioned below the x-axis. Knowing this direction helps us understand not just where the vector ends but the path it follows from the origin.

Pythagorean Theorem

The Pythagorean Theorem is a principle of geometry that relates the sides of a right triangle. It states that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

• \(a^2 + b^2 = c^2\)

In vector analysis, we can use this theorem to determine the magnitude of a vector. Here, the vector components along the axes form the two shorter sides of the triangle. Then, the vector itself becomes the hypotenuse. By applying the Pythagorean Theorem, you solve for the vector's total length.
For example, if we have components \(V_x = 7.80\) and \(V_y = -6.40\), we plug these into the theorem's formula:
\[(7.80)^2 + (-6.40)^2 = (\overrightarrow{\mathbf{V}})^2\] Solving this equation helps demonstrate that the vector's magnitude, as calculated by its components, is consistent and precise.