Question

Find \((a) \mathbf{u} \times \mathbf{v},(b) \mathbf{v} \times \mathbf{u},\) and \((\mathbf{c}) \mathbf{v} \times \mathbf{v}\). $$ \begin{array}{l} \mathbf{u}=\langle 7,3,2\rangle \\ \mathbf{v}=\langle 1,-1,5\rangle \end{array} $$

Short answer

The cross product results are \((a) \mathbf{u} \times \mathbf{v} = \langle 17,-29,-10\rangle, (b) \mathbf{v} \times \mathbf{u} = \langle -17,29,10\rangle\), and \(c) \mathbf{v} \times \mathbf{v} = 0\).

Step-by-step solution

Calculating \(\mathbf{u} \times \mathbf{v}\)

The cross product of vectors \(\mathbf{u} = \langle 7,3,2\rangle\) and \(\mathbf{v} = \langle 1,-1,5\rangle\) can be calculated with the determinant of a 3×3 matrix. The matrix is defined like this:\[\begin{bmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \u_1 (7) & u_2 (3) & u_3 (2) \v_1 (1) & v_2 (-1) & v_3 (5) \\end{bmatrix}\]Calculate the determinant of this matrix to find \(\mathbf{u} \times \mathbf{v}\).

Calculating \(\mathbf{v} \times \mathbf{u}\)

We can reapply the same process as in the previous step, but reversing the order of the vectors. We know cross product is not commutative, so \(\mathbf{v} \times \mathbf{u}\) would be negative of \(\mathbf{u} \times \mathbf{v}\).

Calculating \(\mathbf{v} \times \mathbf{v}\)

The cross product of any vector with itself is always the zero vector. This concept holds true for any vector in any number of dimensions. So, \(\mathbf{v} \times \mathbf{v}\) is \(0\).

Key concepts

Cross Product Calculation

Understanding the cross product calculation is crucial for students delving into the world of vectors and three-dimensional space. This operation takes two vectors and produces a third vector that is perpendicular to both original vectors. The length of the resulting vector reflects the area of the parallelogram formed by the original vectors.

For vectors \textbf{u} = \(7,3,2\) and \textbf{v} = \(1,-1,5\), we can calculate the cross product using the matrix method. First, we create a matrix with three rows; the top row contains the unit vectors \textbf{i}, \textbf{j}, and \textbf{k}, which represent the x, y, and z axes, respectively. The next two rows contain the corresponding components of vectors \textbf{u} and \textbf{v}. We then find the determinant of this matrix to get the cross product: \textbf{u} \times \textbf{v}.

Remember, it’s vital to maintain the order of the vectors in the calculation since the cross product is not commutative—switching the order will reverse the sign of the result, yielding \textbf{v} \times \textbf{u} = -(\textbf{u} \times \textbf{v}). This property highlights the importance of careful and precise calculation in vector algebra.

Vectors in Calculus

In calculus, vectors play an indispensable role in understanding and solving problems related to motion, forces, and fields within three-dimensional space. A vector in this context is not just a set of numbers; it's a quantity with both magnitude and direction. This is why, when we talk about vectors in calculus, we're often interested in operations that respect these properties, such as vector addition, scalar multiplication, dot product, and cross product.

Vectors provide a way to navigate three dimensions by allowing for precise and directionally aware computations. For example, when dealing with velocity and acceleration in physics, vectors help us analyze the movement of an object in space. The cross product, specifically, is used to find a vector that is orthogonal to the plane defined by two given vectors, which can help in determining the rotational forces or the orientation of an object in 3D space.

Developing an understanding of vectors and their operations is a fundamental step in calculus, as it equips students with the tools necessary for tackling complex problems in physics, engineering, and mathematics.

Determinant of a Matrix

The determinant of a matrix is a scalar value that can be computed from the elements of a square matrix and offers critical information about the matrix, such as whether it is invertible or not. In the context of vectors and the cross product, the determinant is used to calculate the magnitude and direction of the vector produced by the cross product.

For a 3x3 matrix, the determinant is found by expanding the matrix using the rule of Sarrus or the more general Laplace's formula. While the process can seem complex at first glance, it's a systematic method that, once mastered, becomes a standard tool in a student's skill set. The determinant gives us the volume scaling factor when the vectors are interpreted as sides of a parallelogram in space. If the determinant is zero, it indicates that the vectors are linearly dependent, and the parallelogram collapses to a line or a point.

When considering the cross product, we utilize the components of the vectors involved and construct a matrix with these components. We calculate its determinant to find the resulting vector that is perpendicular to the original vectors. This process is beautifully interconnected with the geometric interpretation of vectors, and it underpins much of the work done in vector calculus and physics.