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 \(\vec{v}\). \(\vec{u}=\vec{i}, \quad \vec{v}=\vec{j}\)

Short answer

The cross product is \( \vec{k} \), which is orthogonal to both given vectors.

Step-by-step solution

Identify the Vectors

We are given two vectors, \( \vec{u} = \vec{i} \) and \( \vec{v} = \vec{j} \). Here, \( \vec{i} \) and \( \vec{j} \) are the unit vectors along the x-axis and y-axis, respectively.

Compute the Cross Product

The cross product \( \vec{u} \times \vec{v} \) is computed by using the determinant of a matrix with the unit vector components. The cross product of two vectors \( \vec{a} = a_1\vec{i} + a_2\vec{j} + a_3\vec{k} \) and \( \vec{b} = b_1\vec{i} + b_2\vec{j} + b_3\vec{k} \) is \( \vec{a} \times \vec{b} = (a_2b_3 - a_3b_2)\vec{i} - (a_1b_3 - a_3b_1)\vec{j} + (a_1b_2 - a_2b_1)\vec{k} \). For our vectors, \( a_1 = 1, a_2 = 0, a_3 = 0 \), and \( b_1 = 0, b_2 = 1, b_3 = 0 \).

Key concepts

Orthogonal Vectors

When we talk about orthogonal vectors, we're referring to vectors that are perpendicular to each other. This concept is significant in vector mathematics. If two vectors are orthogonal, their dot product equals zero. For example, let's consider two vectors \(\vec{a}\) and \(\vec{b}\):* If \(\vec{a} \cdot \vec{b} = 0\), we know they are orthogonal.In the context of the cross product, the result of a vector cross product, \(\vec{u} \times \vec{v}\), will always produce a vector that is orthogonal (or perpendicular) to both \(\vec{u}\) and \(\vec{v}\).In our example with vectors \(\vec{i}\) and \(\vec{j}\), the cross product is a vector in the z-direction, \(\vec{k}\). Since \(\vec{k}\) is perpendicular to \(\vec{i}\) and \(\vec{j}\), it confirms the orthogonality.

Unit Vectors

Unit vectors are essential building blocks in vector mathematics. They are vectors with a magnitude of 1, used to indicate direction alone. The unit vectors \(\vec{i}, \vec{j},\) and \(\vec{k}\) represent the standard basis vectors in three-dimensional space:* \(\vec{i}\) points in the x-direction,* \(\vec{j}\) points in the y-direction,* \(\vec{k}\) points in the z-direction.These vectors simplify calculations and are very useful when working with the cross product. For instance,.When you multiply two unit vectors like \(\vec{i}\) and \(\vec{j}\), the resulting vector \(\vec{k}\) is also a unit vector as it maintains the property of having a magnitude of 1.Unit vectors serve as a reference system for describing any vector's direction in space.

Determinant in Cross Product

The cross product of two vectors can be computed using a determinant. This is a common technique that helps determine the vector that results from the cross product.Let's recall, the formula for the cross product of vectors \(\vec{a}\) and \(\vec{b}\):1. Arrange them in the form of a matrix with three rows: * First row: Unit vectors \(\vec{i}, \vec{j}, \vec{k}\) * Second row: Components of \(\vec{a}\) * Third row: Components of \(\vec{b}\)2. Compute the determinant of this 3x3 matrix. The determinant gives us a new vector that is perpendicular to both \(\vec{a}\) and \(\vec{b}\). * The determinant is calculated as: \[ \vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \]Where the resulting vector's components are derived from the subtraction of products of minor matrices.In our example, the simple case of \(\vec{u} = \vec{i}\) and \(\vec{v} = \vec{j}\) led to the vector \(\vec{k}\).