Problem 2 - University Calculus: Early Transcendentals

Question

Find the length and direction (when defined) of \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\). $$\mathbf{u}=2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{v}=-\mathbf{i}+\mathbf{j}$$

Step-by-Step Solution

Answer

\(\mathbf{u} \times \mathbf{v} = 5\mathbf{k}\) with length 5, direction positive \(z\)-axis. \(\mathbf{v} \times \mathbf{u} = -5\mathbf{k}\) with length 5, direction negative \(z\)-axis.

Step 1: Understand the Cross Product

The cross product of two vectors \(\mathbf{u} \times \mathbf{v}\) can be found using the determinant of a matrix formed with the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) and the components of \( \mathbf{u} \) and \( \mathbf{v} \). The resulting vector is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \).

Step 2: Set up the Determinant for \(\mathbf{u} \times \mathbf{v}\)

Construct the matrix for \(\mathbf{u} \times \mathbf{v}\):\[\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \2 & 3 & 0 \-1 & 1 & 0 \end{vmatrix}\]The third component is zero because both \(\mathbf{u}\) and \(\mathbf{v}\) are in the \(xy\)-plane.

Step 3: Calculate the Cross Product \(\mathbf{u} \times \mathbf{v}\)

Expand the determinant:\[\mathbf{u} \times \mathbf{v} = \mathbf{i}(3 \times 0 - 0 \times 1) - \mathbf{j}(2 \times 0 - 0 \times (-1)) + \mathbf{k}(2 \times 1 - 3 \times (-1))\]Simplifying this gives:\[\mathbf{u} \times \mathbf{v} = 0\mathbf{i} - 0\mathbf{j} + 5\mathbf{k} = 5\mathbf{k}\]

Step 4: Determine the Length of \(\mathbf{u} \times \mathbf{v}\)

The length of the vector \(\mathbf{u} \times \mathbf{v} = 5 \mathbf{k}\) is calculated as:\[|\mathbf{u} \times \mathbf{v}|= \sqrt{0^2 + 0^2 + 5^2} = 5\]

Step 5: Analyze the Direction of \(\mathbf{u} \times \mathbf{v}\)

The direction of \(\mathbf{u} \times \mathbf{v}\) is along the positive \(z\)-axis, as the result is \(5\mathbf{k}\).

Step 6: Set up the Determinant for \(\mathbf{v} \times \mathbf{u}\)

Construct the matrix for \(\mathbf{v} \times \mathbf{u}\):\[\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \-1 & 1 & 0 \2 & 3 & 0 \end{vmatrix}\]

Step 7: Calculate the Cross Product \(\mathbf{v} \times \mathbf{u}\)

Expand the determinant:\[\mathbf{v} \times \mathbf{u} = \mathbf{i}(1 \times 0 - 0 \times 3) - \mathbf{j}((-1) \times 0 - 0 \times 2) + \mathbf{k}((-1)\times 3 - 1 \times 2)\]Simplifies to:\[\mathbf{v} \times \mathbf{u} = 0\mathbf{i} - 0\mathbf{j} - 5\mathbf{k} = -5\mathbf{k}\]

Step 8: Determine the Length of \(\mathbf{v} \times \mathbf{u}\)

The length of the vector \(\mathbf{v} \times \mathbf{u} = -5 \mathbf{k}\) is:\[|\mathbf{v} \times \mathbf{u}|= \sqrt{0^2 + 0^2 + (-5)^2} = 5\]

Step 9: Analyze the Direction of \(\mathbf{v} \times \mathbf{u}\)

The direction of \(\mathbf{v} \times \mathbf{u}\) is along the negative \(z\)-axis, as it results in \(-5\mathbf{k}\).

Key Concepts

Vector Mathematics

Vectors are fundamental components in mathematics and physics, representing magnitude and direction. They are pivotal in describing quantities that involve both these characteristics such as force, velocity, and displacement. In vector mathematics, operations like addition, subtraction, and multiplication (both dot and cross product) are used to solve various problems.

• Addition: Vectors are added by summing their corresponding components.
• Subtraction: Similar to addition, but involves subtracting the corresponding components.
• Multiplication: Can be done via the dot product or cross product depending on whether a scalar or vector result is needed.

The cross product of two vectors is unique in a sense, as it results in another vector that is perpendicular to the plane formed by the original vectors. Unlike the dot product, which results in a scalar, the cross product provides insights into the spatial orientation of vectors, and is a key tool in 3D vector problems.

Determinants

Determinants are mathematical tools used to solve systems of linear equations, calculate areas and volumes, and find transformation properties in geometry. For vectors, determinants provide a structured way to find the cross product.

• To calculate the cross product using determinants, a matrix framework is set up using the basis vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).
• The components of the original vectors are placed in this framework to form a square matrix.
• Then, the determinant of this matrix is computed to find the cross product.

For the vectors \( \mathbf{u} \) and \( \mathbf{v} \) as given in the exercise, their determinants are set up with zero in the \( z \)-component, revealing their confinement to the \( xy \)-plane.
Determinants offer a neat encapsulation of the relationships between vector components, allowing a deeper understanding of spatial interactions, especially crucial when dealing with 3D problems.

3D Vectors

Three-dimensional vectors extend the principles of vector mathematics into the realm of three variables, usually represented by the coordinates \( x, y, \) and \( z \). These vectors are essential in depicting physics problems involving spatial relations such as those in engineering and computer graphics.

• 3D vectors have three components, which makes operations like dot and cross product significant as they determine geometrical relationships.
• The cross product, in particular, results in a vector that is perpendicular to the plane containing the original two vectors, thus crucial in determining orientation.
• Given two vectors in 3D space, their cross product tells us about their mutual alignment and helps define volume using the resulting vector's magnitude.

In the context of this exercise, operations were simplified since both vectors had no \( z \)-components, lying in the \( xy \)-plane. However, the cross product result \( 5\mathbf{k} \) for \( \mathbf{u} \times \mathbf{v} \) and \( -5\mathbf{k} \) for \( \mathbf{v} \times \mathbf{u} \) firmly illustrates their perpendicularity to the plane, showcasing the intrinsic connection between vectors in 3D.