Problem 15 - APEX Calculus

Question

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\vec{i}, \quad \vec{v}=\vec{k}\)

Step-by-Step Solution

Answer

The cross product is \(-\vec{j}\) and it's orthogonal to \(\vec{i}\) and \(\vec{k}\).

Step 1: Set Up the Cross Product

The cross product of two vectors \( \vec{u} \) and \( \vec{v} \) is denoted as \( \vec{u} \times \vec{v} \). For the vectors \( \vec{u} = \vec{i} \) and \( \vec{v} = \vec{k} \), we set up the formula \( \vec{u} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \ 1 & 0 & 0 \ 0 & 0 & 1 \end{vmatrix} \).

Step 2: Compute the Determinant

Calculate the determinant to find \( \vec{u} \times \vec{v} \). For the vectors: the determinant is \( \vec{i}(0 \cdot 1 - 0 \cdot 0) - \vec{j}(1 \cdot 1 - 0 \cdot 0) + \vec{k}(1 \cdot 0 - 0 \cdot 0) = -\vec{j} \). So, \( \vec{u} \times \vec{v} = -\vec{j} \).

Step 3: Check Orthogonality to \( \vec{u} \)

To check if a vector is orthogonal to another vector, their dot product should be zero. Calculate \( \vec{u} \cdot (-\vec{j}) = \vec{i} \cdot (-\vec{j}) = 0(-1) = 0 \). The dot product is zero, confirming orthogonality.

Step 4: Check Orthogonality to \( \vec{v} \)

Similarly, check the orthogonality with \( \vec{v} = \vec{k} \). Calculate \( \vec{k} \cdot (-\vec{j}) = 0(-1) = 0 \). Their dot product is zero, so \( \vec{u} \times \vec{v} \) is also orthogonal to \( \vec{v} \).

Key Concepts

Understanding Orthogonal Vectors

Orthogonal vectors are vectors that are perpendicular to each other. This means that they form a right angle (90 degrees) with each other in space. When vectors are orthogonal, they have a special property: their dot product is zero. To better understand this, imagine two arrows pointing in different directions. If they do not "pull" on each other at all, they are orthogonal. This lack of interaction translates to a zero dot product.
When dealing with vectors, it's important to note that if you're working in a three-dimensional space, a vector can be orthogonal to multiple other vectors. This is precisely what happens when we compute a cross product; the resulting vector is orthogonal to both of the original vectors involved in the operation.

Determinant Calculation in Cross Products

The determinant plays a crucial role in calculating the cross product of two vectors. It is a mathematical tool used to compute an area or volume, depending on the context. In the case of vectors, it helps determine the vector perpendicular to two given vectors.
The cross product is visualized as a determinant setup. For example, for vectors \( \vec{u} = \vec{i} \) and \( \vec{v} = \vec{k} \), the determinant is set up using a 3x3 matrix including the unit vectors \( \vec{i} \), \( \vec{j} \), and \( \vec{k} \).

• The first row consists of the unit vectors.
• The second row contains the components of the first vector, \( \vec{u} \).
• The third row contains the components of the second vector, \( \vec{v} \).

This setup allows us to apply the cross product formula. Solving this determinant gives us the resultant orthogonal vector. This method ensures that the mathematical process respects the geometry of the problem.

The Dot Product - Checking Orthogonality

The dot product is a vector operation that helps us determine if two vectors are orthogonal. Mathematically, the dot product is the sum of the products of the corresponding components of two vectors. If the dot product is zero, the vectors are orthogonal.
Let's look at why this happens. The dot product can be defined as:

• The product of the magnitudes of the two vectors and the cosine of the angle between them.

Mathematically, for two vectors \( \vec{a} \) and \( \vec{b} \), it is expressed as \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \).
For orthogonal vectors, \( \theta = 90 \) degrees, and thus \( \cos(90) = 0 \). Hence, the dot product becomes zero, confirming their orthogonality. This allows us to easily verify when vectors are perpendicular by simply looking at their dot product outcome.

Exploring Vector Operations

Vector operations like the cross and dot products are fundamental in understanding the spatial relationships between vectors. These operations are used extensively in physics and engineering.
The cross product, represented by the symbol \( \times \), generates a vector that is perpendicular to the plane formed by two input vectors. It represents a rotational aspect and can be used to compute torque or angular momentum in physical contexts.

• The result of a cross product is a vector.
• The direction of this vector is determined by the right-hand rule, ensuring a standardized orientation.

On the other hand, the dot product is a scalar result and is useful in determining the angle between two vectors or finding the projection of one vector onto another.
Both operations are crucial for solving problems involving forces, motion, and mechanics. Understanding how to compute and apply these operations helps in many fields that require spatial calculations, showing the power and utility of vector algebra.