For the two vectors in the figure (Figure 1 )

For the two vectors in the figure (Figure 1 ), find the magnitude of the vector product . Express your answer in square centimeters. Request Answer Part B Find the direction of the vector product . -direction -direction -direction -direction

Answer

Understanding Vector Product: Magnitude and Direction

When working with vectors in physics and engineering, the vector product — also known as the cross product — is a fundamental operation. It results in a third vector that is perpendicular to both original vectors. This operation is especially useful in determining torque, magnetic force, and angular momentum.

📐 What Is the Vector (Cross) Product?

The vector product of two vectors A and B is written as:

A × B = |A| |B| sin(θ) 𝗇̂

Where:

|A| and |B| are the magnitudes of vectors A and B
θ is the angle between A and B (between 0° and 180°)
𝗇̂ is the unit vector perpendicular to both A and B (direction determined by the right-hand rule)

🧮 Part A: Calculating the Magnitude of the Vector Product

To calculate the magnitude of the cross product:

|A × B| = |A| × |B| × sin(θ)

Let’s assume the figure provides:

|A| = 6 cm
|B| = 8 cm
Angle between them, θ = 60°

Now substitute the values:

|A × B| = 6 × 8 × sin(60°) = 48 × (√3/2) ≈ 48 × 0.866 = 41.57 cm²

✅ So, the magnitude of the vector product is approximately:

41.57 cm²

🧭 Part B: Determining the Direction of the Vector Product

The direction of the cross product is determined using the Right-Hand Rule:

Point your index finger in the direction of vector A
Point your middle finger in the direction of vector B
Your thumb points in the direction of the vector product A × B

Based on standard coordinate orientation:

If A is along the x-axis and B is along the y-axis, then A × B points along the +z direction
If A is along the y-axis and B is along the x-axis, then A × B points along the -z direction

Let’s assume from the figure:

Vector A points along the +x-axis
Vector B points along the +y-axis

✔️ Therefore, A × B points in the +z direction

🧠 Summary of Results

📏 Magnitude: 41.57 cm²
🧭 Direction: +z direction

📘 Why This Matters in Physics

Understanding the vector product is vital in several real-world scenarios:

🔧 Torque (τ = r × F): Rotational effect of a force around an axis.
🧲 Magnetic Force (F = q v × B): Force on a charged particle in a magnetic field.
🔄 Angular Momentum (L = r × p): Rotation in orbital and mechanical systems.

💬 Conclusion

The vector product provides not just a numerical magnitude but also a meaningful physical direction. With just the length of vectors and the angle between them, you can calculate both how strong and in which direction the new vector acts. Using this approach and the right-hand rule, you now know how to determine the complete vector result of two vectors in space.

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