Question

Find the cross products \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\) for the following vectors \(\mathbf{u}\) and \(\mathbf{v}\) $$\mathbf{u}=2 \mathbf{i}-10 \mathbf{j}+15 \mathbf{k}, \mathbf{v}=0.5 \mathbf{i}+\mathbf{j}-0.6 \mathbf{k}$$

Short answer

Question: Calculate the cross products of the given vectors: $$\mathbf{u}=2 \mathbf{i}-10 \mathbf{j}+15 \mathbf{k}, \mathbf{v}=0.5 \mathbf{i}+\mathbf{j}-0.6 \mathbf{k}$$ Answer: The cross products of the given vectors are: $$\mathbf{u} \times \mathbf{v} = -9 \mathbf{i} + 8.7 \mathbf{j} + 7 \mathbf{k}$$ $$\mathbf{v} \times \mathbf{u} = 9 \mathbf{i} - 6.3 \mathbf{j} - 7 \mathbf{k}$$

Step-by-step solution

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

Start by calculating the cross product of the given vectors \(\mathbf{u}\) and \(\mathbf{v}\) using the formula: $$(\mathbf{u} \times \mathbf{v})_i= \begin{vmatrix} u_j & u_k \\ v_j & v_k \end{vmatrix}\mathbf{i}$$ $$(\mathbf{u} \times \mathbf{v})_j= - \begin{vmatrix} u_i & u_k \\ v_i & v_k \end{vmatrix}\mathbf{j}$$ $$(\mathbf{u} \times \mathbf{v})_k= \begin{vmatrix} u_i & u_j \\ v_i & v_j \end{vmatrix}\mathbf{k}$$ So, the cross product \(\mathbf{u} \times \mathbf{v}\) would be: $$\begin{aligned}(\mathbf{u} \times \mathbf{v}) &= (-10 \times (-0.6) - 15 \times 1) \mathbf{i} - (2 \times (-0.6) - 15 \times 0.5) \mathbf{j} + (2 \times 1 - (-10) \times 0.5) \mathbf{k}\\ &= (6-15) \mathbf{i} - (-1.2 - 7.5) \mathbf{j} + (2 + 5) \mathbf{k}\\ &= -9 \mathbf{i} + 8.7 \mathbf{j} + 7 \mathbf{k}\end{aligned}$$

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

Now, we calculate the cross product \(\mathbf{v} \times \mathbf{u}\) using the same formula as before: $$(\mathbf{v} \times \mathbf{u})_i= \begin{vmatrix} v_j & v_k \\ u_j & u_k \end{vmatrix}\mathbf{i}$$ $$(\mathbf{v} \times \mathbf{u})_j= - \begin{vmatrix} v_i & v_k \\ u_i & u_k \end{vmatrix}\mathbf{j}$$ $$(\mathbf{v} \times \mathbf{u})_k= \begin{vmatrix} v_i & v_j \\ u_i & u_j \end{vmatrix}\mathbf{k}$$ So, the cross product \(\mathbf{v} \times \mathbf{u}\) would be: $$\begin{aligned}(\mathbf{v} \times \mathbf{u}) &= (1 \times 15 - (-0.6) \times (-10)) \mathbf{i} - (0.5 \times 15 - 2 \times (-0.6)) \mathbf{j} + (0.5 \times (-10) - 1 \times 2) \mathbf{k}\\ &= (15-6) \mathbf{i} - (7.5 - 1.2) \mathbf{j} + (-5 - 2) \mathbf{k}\\ &= 9 \mathbf{i} - 6.3 \mathbf{j} - 7 \mathbf{k}\end{aligned}$$ Therefore, the cross products \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\) are: $$\mathbf{u} \times \mathbf{v} = -9 \mathbf{i} + 8.7 \mathbf{j} + 7 \mathbf{k}$$ $$\mathbf{v} \times \mathbf{u} = 9 \mathbf{i} - 6.3 \mathbf{j} - 7 \mathbf{k}$$

Key concepts

Vectors

A vector is a fundamental concept in mathematics and physics, representing both a direction and a magnitude. Vectors are typically denoted using boldface letters such as \( \mathbf{u} \) or \( \mathbf{v} \). They can exist in any dimension, but in the context of 3D space, they have three components, each corresponding to one of the three axes (\( \mathbf{i}, \mathbf{j}, \mathbf{k} \)).

Consider a vector \( \mathbf{u} = 2 \mathbf{i} - 10 \mathbf{j} + 15 \mathbf{k} \). Here, 2 is the coefficient of \( \mathbf{i} \), meaning the vector extends 2 units in the x-direction. Similarly, it extends -10 units in the y-direction and 15 units in the z-direction. This vector can be thought of as an arrow pointing in 3D space, starting from the origin and heading toward the point (2, -10, 15).

• Magnitude: The length of a vector. It can be computed using the formula \( \sqrt{x^2 + y^2 + z^2} \) for a 3-dimensional vector.
• Direction: The orientation of the vector in space. The direction can be represented as a unit vector (a vector with a magnitude of 1).

Vectors are crucial for representing physical quantities like velocity, force, and acceleration, which have both magnitude and direction.

Determinants

Determinants are a special number that can be calculated from a square matrix. They are fundamental in linear algebra and come up often when solving for properties of vectors, such as in the calculation of cross products.

In the context of vectors and cross products, determinants represent the 2x2 matrices which calculate the components of the result vector. For instance, when finding \( \mathbf{u} \times \mathbf{v} \), the determinant provides the coefficient for each vector component (\( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \)). The formula to compute a 2x2 determinant is:

\[ \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc \]

For example, when calculating the \( \mathbf{j} \)-component of the cross product \( \mathbf{u} \times \mathbf{v} \), we use the determinant

\[ - \begin{vmatrix} 2 & 15 \ 0.5 & -0.6 \end{vmatrix} \]

This determinant evaluates to \(- (2 \times -0.6 - 15 \times 0.5) \), simplifying to the contribution towards the \( \mathbf{j} \) part of the cross product vector.

Vector Calculus

Vector calculus is a branch of mathematics that deals with differentiation and integration of vector fields, primarily in 2D and 3D. A crucial operation in vector calculus is the cross product, which is used for finding a vector perpendicular to two given vectors in space.

The cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \), denoted as \( \mathbf{u} \times \mathbf{v} \), is only defined in three dimensions and follows specific rules.

• The result of a cross product is another vector, which is orthogonal (perpendicular) to both of the original vectors.
• Its magnitude can be interpreted as the area of the parallelogram formed by \( \mathbf{u} \) and \( \mathbf{v} \).
• The direction of the resulting vector is given by the right-hand rule: if you point your index finger along \( \mathbf{u} \) and your middle finger along \( \mathbf{v} \), your thumb points in the direction of \( \mathbf{u} \times \mathbf{v} \).

An essential property of the cross product is antisymmetry, meaning \( \mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u}) \). This property is clearly reflected in the problem's step-by-step solution where \( \mathbf{u} \times \mathbf{v} = -9 \mathbf{i} + 8.7 \mathbf{j} + 7 \mathbf{k} \) and \( \mathbf{v} \times \mathbf{u} = 9 \mathbf{i} - 6.3 \mathbf{j} - 7 \mathbf{k} \).

Understanding these concepts is fundamental for applications in physics, engineering, and computer graphics.