Problem 16 - 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 \(\vec{v}\). \(\vec{u}=\vec{j}, \quad \vec{v}=\vec{k}\)

Step-by-Step Solution

Answer

The cross product \( \vec{u} \times \vec{v} = \vec{i} \) and it is orthogonal to both \( \vec{u} \) and \( \vec{v} \).

Step 1: Understand the Vectors

The vectors given are \( \vec{u} = \vec{j} \) and \( \vec{v} = \vec{k} \). These are the standard basis vectors in 3D, where \( \vec{j} = [0, 1, 0] \) and \( \vec{k} = [0, 0, 1] \).

Step 2: Set up the Cross Product Formula

The cross product \( \vec{u} \times \vec{v} \) can be calculated using the determinant of a matrix consisting of \( \vec{i}, \vec{j}, \vec{k} \) as the first row, and the components of \( \vec{u} \) and \( \vec{v} \) as the second and third rows, respectively. \[\vec{u} \times \vec{v} = \begin{vmatrix}\vec{i} & \vec{j} & \vec{k} \0 & 1 & 0 \0 & 0 & 1\end{vmatrix}.\]

Step 3: Compute the Determinant

The determinant gives us the cross product: \[\vec{u} \times \vec{v} = \vec{i}(1 \cdot 1 - 0 \cdot 0) - \vec{j}(0 \cdot 1 - 0 \cdot 0) + \vec{k}(0 \cdot 0 - 1 \cdot 0) = \vec{i}.\] So, \( \vec{u} \times \vec{v} = \vec{i} = [1, 0, 0]. \)

Step 4: Show Orthogonality to \( \vec{u} \)

A vector is orthogonal to another if their dot product is zero. Compute the dot product of \( \vec{u} \times \vec{v} = \vec{i} \) with \( \vec{u} = \vec{j} \):\[[1, 0, 0] \cdot [0, 1, 0] = 1\cdot0 + 0\cdot1 + 0\cdot0 = 0.\]Since the dot product is 0, \( \vec{u} \times \vec{v} \) is orthogonal to \( \vec{u} \).

Step 5: Show Orthogonality to \( \vec{v} \)

Similarly, compute the dot product of \( \vec{u} \times \vec{v} = \vec{i} \) with \( \vec{v} = \vec{k} \):\[[1, 0, 0] \cdot [0, 0, 1] = 1\cdot0 + 0\cdot0 + 0\cdot1 = 0.\]The dot product is 0, which means \( \vec{u} \times \vec{v} \) is orthogonal to \( \vec{v} \).

Key Concepts

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle. This is crucial because, when dealing with vector spaces, orthogonality implies independence. The defining property of orthogonal vectors is that their dot product equals zero. This means there is no overlap or projection in the direction of each vector on the other.
In our exercise, after computing the cross product of vectors \(\vec{u}\) and \(\vec{v}\), we get \(\vec{i} = [1, 0, 0]\). We prove orthogonality by showing the dot products \([1, 0, 0] \cdot [0, 1, 0] = 0\) and \([1, 0, 0] \cdot [0, 0, 1] = 0\). These calculations confirm that the resulting vector \(\vec{i}\) is orthogonal to both \(\vec{j}\) and \(\vec{k}\).
Orthogonal vectors are essential in many areas of mathematics and physics, where they often simplify complex problems by reducing dimensions in calculations.

Determinant Calculation

The determinant plays a vital role in finding the cross product of two vectors. We're using this method specifically to leverage the properties of determinants in operations involving 3D vectors. In a symbolic sense, a determinant helps deal with issues like matrix singularity and vector scaling, giving us the necessary tools to find orthogonal vectors from initial inputs.
\[\vec{u} \times \vec{v} = \begin{vmatrix}\vec{i} & \vec{j} & \vec{k} \ 0 & 1 & 0 \ 0 & 0 & 1\end{vmatrix}\]
In this format, we use the standard basis axes \(\vec{i}, \vec{j},\) and \(\vec{k}\) along with the components of \(\vec{u}\) and \(\vec{v}\). The cross product vector \(\vec{i}\) arises from performing the determinant operation: \(\vec{i}(1 \cdot 1 - 0 \cdot 0) - \vec{j}(0 \cdot 1 - 0 \cdot 0) + \vec{k}(0 \cdot 0 - 1 \cdot 0)\).
Determinants are a powerful tool in linear algebra because they provide a scalar value that represents the volume scaling factor of vectors and matrices, further showcased in multidimensional geometry applications.

Vector Operations

Vector operations encompass computations like addition, subtraction, dot product, and cross product. They are pivotal in fields ranging from engineering to computer graphics and beyond. Understanding these operations provides foundational insight for more advanced concepts like vector fields and transformations.
The cross product, specifically, is a vector operation used in three-dimensional space to determine a vector perpendicular to two given vectors. It results in a vector that is orthogonal to the original vectors, as demonstrated here with \(\vec{u} = \vec{j}\) and \(\vec{v} = \vec{k}\). By calculating \(\vec{u} \times \vec{v}\), we implement the determinant method we just discussed to verify the outcome as \(\vec{i}\), which reveals deep geometrical insights.
These operations allow us to manipulate vectors in a manner consistent with physical and mathematical theories, enabling us to tackle problems effectively within physics, engineering, computer science, and many other disciplines.