Problem 39 - Fundamentals of Physics

Question

Vector \(\vec{A}\) has a magnitude of \(6.00\) units, vector \(\vec{B}\) has a mag. nitude of \(7.00\) units, and \(\vec{A} \cdot \vec{B}\) has a value of \(14.0\). What is the angle between the directions of \(\vec{A}\) and \(\vec{B}\) ?

Step-by-Step Solution

Answer

The angle between the vectors is approximately 70.5 degrees.

Step 1: Understand the Dot Product Formula

To find the angle between two vectors \(\vec{A}\) and \(\vec{B}\), we use the dot product formula: \(\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta\), where \(\theta\) is the angle between the vectors. Given, \(\vec{A} \cdot \vec{B} = 14.0\), \(|\vec{A}| = 6.00\), and \(|\vec{B}| = 7.00\).

Step 2: Substitute Known Values into the Formula

Substitute the given values into the dot product formula: \[ 14.0 = 6.00 \times 7.00 \times \cos \theta\]

Step 3: Simplify the Equation

Simplify the right side of the equation:\[ 14.0 = 42 \times \cos \theta\]

Step 4: Solve for Cosine of Theta

Rearrange the equation to solve for \(\cos \theta\):\[ \cos \theta = \frac{14.0}{42}\]

Step 5: Calculate the Cosine Value

Calculate the cosine value:\[ \cos \theta = 0.3333\]

Step 6: Determine Theta Using Inverse Cosine

To find \(\theta\), apply the inverse cosine function:\[ \theta = \cos^{-1}(0.3333)\]Calculating this gives \(\theta \approx 70.5\degree\).

Key Concepts

The Dot Product: A Fundamental Building Block

The dot product is a calculation that combines two vectors and returns a scalar. It's an essential concept in understanding relationships between vectors, especially when determining angles. When you take the dot product of two vectors \( \vec{A} \) and \( \vec{B} \), represented as \( \vec{A} \cdot \vec{B} \), you're essentially calculating \( |vec{A}| |vec{B}| \cos \theta \), where \( \theta \) is the angle between the two vectors.

• If the dot product is positive, the vectors point in similar directions.
• If it's negative, the vectors point in opposite directions.
• If it's zero, the vectors are perpendicular.

Therefore, by using the dot product and knowing the magnitudes of the vectors, you can easily solve for the angle \( \theta \), which gives you an insight into how these two vectors are oriented in space.

Magnitude of a Vector: Understanding Its Length

The magnitude of a vector, often denoted as \( |vec{A}| \), represents its 'length' or 'size.' It's similar to how long a line is in geometry. For any vector \( \vec{A} = (x, y, z) \), the magnitude is calculated as \( |vec{A}| = \sqrt{x^2 + y^2 + z^2} \).
Magnitude is crucial since it tells us about the scale of a vector. Having the magnitude helps us use formulas like the dot product effectively, especially when seeking the angle between vectors. In our problem, knowing that \( |vec{A}| = 6.00 \) units and \( |vec{B}| = 7.00 \) units allows us to substitute these into the formula \( \vec{A} \cdot \vec{B} = |vec{A}| |vec{B}| \cos \theta \).
This substitution helps in simplifying the equation as we proceed to solve for the angle.

Inverse Cosine: Finding the Angle

The inverse cosine, often denoted as \( \cos^{-1} \), is a mathematical operation that helps us find an angle when the cosine value is known. It's the reverse of finding the cosine of an angle. When you have a cosine value, \( \cos \theta \), the inverse cosine function gives back the angle \( \theta \).
Applying the inverse cosine is straightforward. In the context of this problem, after finding \( \cos \theta = 0.3333 \), we use \( \theta = \cos^{-1}(0.3333) \).
The calculated angle for our specific vectors \( \vec{A} \) and \( \vec{B} \), is approximately \( 70.5^\circ \). Knowing how to use the inverse cosine is critical in problems involving angles, as it connects the dot product directly to the spatial orientation of vectors.