Problem 20 - APEX Calculus

Question

The magnitudes of vectors \(\vec{u}\) and \(\vec{v}\) in \(\mathbb{R}^{3}\) are given, along with the angle \(\theta\) between them. Use this information to find the magnitude of \(\vec{u} \times \vec{v}\). \(\|\vec{u}\|=3, \quad\|\vec{v}\|=7, \quad \theta=\pi / 2\)

Step-by-Step Solution

Answer

The magnitude of \(\vec{u} \times \vec{v}\) is 21.

Step 1: Understand Vector Cross Product

The vector cross product of two vectors \(\vec{u}\) and \(\vec{v}\) in \(\mathbb{R}^{3}\) is given by the formula \(\|\vec{u} \times \vec{v}\| = \|\vec{u}\| \cdot \|\vec{v}\| \cdot \sin(\theta)\). The result is a vector that is orthogonal to both \(\vec{u}\) and \(\vec{v}\), with the magnitude calculated by the product of their magnitudes and the sine of the angle between them.

Step 2: Plug in the Known Values

You are given the magnitudes \(\|\vec{u}\|=3\) and \(\|\vec{v}\|=7\), as well as the angle \(\theta = \frac{\pi}{2}\). Substitute these values into the cross product magnitude formula: \(\|\vec{u} \times \vec{v}\| = 3 \cdot 7 \cdot \sin\left(\frac{\pi}{2}\right)\).

Step 3: Calculate the Sine of the Angle

Since \(\theta = \frac{\pi}{2}\), \(\sin\left(\frac{\pi}{2}\right) = 1\). This means the sines simplifies to 1, simplifying the calculation of the cross product magnitude.

Step 4: Compute the Magnitude of the Cross Product

Now perform the multiplication: \(\|\vec{u} \times \vec{v}\| = 3 \cdot 7 \cdot 1 = 21\). Therefore, the magnitude of the cross product \(\|\vec{u} \times \vec{v}\|\) is 21.

Key Concepts

Cross Product

The cross product, also known as the vector product, is an operation used in vector calculus commonly applied to vectors in three-dimensional space, \( \mathbb{R}^{3} \). When you take the cross product of two vectors \( \vec{u} \) and \( \vec{v} \), the result is a third vector that is orthogonal to both \( \vec{u} \) and \( \vec{v} \). This unique feature makes the cross product particularly useful in physics and engineering, especially for finding normals to surfaces.

• **Formula**: The magnitude of the cross product is given by \( \|\vec{u} \times \vec{v}\| = \|\vec{u}\| \cdot \|\vec{v}\| \cdot \sin(\theta) \), where \( \theta \) is the angle between the vectors.
• **Orthogonality**: The resulting vector is always perpendicular to the plane containing \( \vec{u} \) and \( \vec{v} \).
• **Direction**: The direction of the cross product is determined by the right-hand rule, which states that if you point your index finger in the direction of \( \vec{u} \) and your middle finger in the direction of \( \vec{v} \), your thumb will point in the direction of \( \vec{u} \times \vec{v} \).

Applying these principles to our exercise, using the formula \( \|\vec{u} \times \vec{v}\| = 3 \cdot 7 \cdot \sin\left(\frac{\pi}{2}\right) \), we find the magnitude of the cross product to be 21.

Magnitude of Vectors

Understanding the magnitude of vectors is crucial in vector calculus. The magnitude of a vector \( \vec{a} \) in \( \mathbb{R}^{3} \) is essentially the length of the vector. It is calculated using the Euclidean norm, which involves squaring each component of the vector, summing these squares, and then taking the square root of the total.

• **Formula**: For a vector \( \vec{a} = (a_{1}, a_{2}, a_{3}) \), the magnitude is \( \|\vec{a}\| = \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}} \).
• **Use**: The magnitude is a measure of how long the vector is, without any direction.
• **Application**: In our case, we have \( \|\vec{u}\| = 3 \) and \( \|\vec{v}\| = 7 \). These are the given magnitudes of our two vectors.

By knowing the magnitudes of vectors \( \vec{u} \) and \( \vec{v} \), we can calculate other properties, such as the cross product, as demonstrated in this exercise.

Angle Between Vectors in R3

The angle between two vectors in \( \mathbb{R}^{3} \) is an important concept in vector calculus. It involves determining the orientation of one vector relative to another in three-dimensional space. The angle \( \theta \) between vectors \( \vec{u} \) and \( \vec{v} \) is a key factor in many calculations involving vectors.

• **Relation to Cross Product**: For the cross product, the sine of the angle is used: \( \sin(\theta) \).
• **Full Range**: The angle \( \theta \) can range from \( 0 \) to \( \pi \), where \( 0 \) indicates the vectors are parallel, and \( \pi \) indicates they are anti-parallel.
• **Given Information**: In our exercise, \( \theta = \frac{\pi}{2} \), meaning the vectors are perpendicular, and \( \sin\left(\frac{\pi}{2}\right) = 1 \).

This specific angle makes calculations straightforward because the cross product simplifies. The vectors being perpendicular (at a right angle) maximizes the magnitude of the cross product, leading to the outcome of 21.