Problem 2 - Calculus: Early Transcendentals

Question

Find the cross product of \langle 1,0,2\rangle and \(\langle-1,-2,4\rangle .\)

Step-by-Step Solution

Answer

The cross product is \( \langle 4, -6, -2 \rangle \).

Step 1: Understand the Cross Product Formula

The cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) can be found using the formula:\[ \mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle \] where \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \).

Step 2: Identify Vector Components

Identify the components of vectors \( \mathbf{a} \) and \( \mathbf{b} \):\[ \mathbf{a} = \langle 1, 0, 2 \rangle \text{ and } \mathbf{b} = \langle -1, -2, 4 \rangle \]Thus, \( a_1 = 1, a_2 = 0, a_3 = 2 \) and \( b_1 = -1, b_2 = -2, b_3 = 4 \).

Step 3: Calculate Each Component of the Cross Product

Use the formula to calculate each component:- First Component: \( a_2b_3 - a_3b_2 = 0\cdot4 - 2\cdot(-2) = 0 + 4 = 4 \).- Second Component: \( a_3b_1 - a_1b_3 = 2\cdot(-1) - 1\cdot4 = -2 - 4 = -6 \).- Third Component: \( a_1b_2 - a_2b_1 = 1\cdot(-2) - 0\cdot(-1) = -2 + 0 = -2 \).

Step 4: Write the Resulting Cross Product

Combine the calculated components into the resulting vector:\[ \mathbf{a} \times \mathbf{b} = \langle 4, -6, -2 \rangle \]

Key Concepts

Vector Multiplication

Vector multiplication can be performed using two primary operations: the dot product and the cross product. While the dot product results in a scalar value, the cross product produces another vector. The focus here is on the cross product, which is essential in three-dimensional space.

• The cross product of two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), results in a new vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \).
• The formula for the cross product is \( \mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle \).
• This operation is particularly useful in physics and engineering to determine torque, rotational motion, and magnetic forces.

Understanding how to calculate the cross product involves identifying the components along each axis and using the formula. This ensures that the resulting vector points in the correct direction according to the right-hand rule. The magnitude of this vector can be computed by determining the product of the magnitudes of the original vectors and the sine of the angle between them.

Vector Calculus

Vector calculus is a branch of mathematics focused on differentiation and integration of vector fields. It extends the concepts from traditional calculus to multiple dimensions, making it an essential tool in physics and engineering.

• Vector calculus operations include gradient, divergence, curl, and the Laplacian—each useful for describing various physical phenomena, like fluid flow and electromagnetic fields.
• The cross product is often a part of vector calculus, as it helps in calculating the curl of a vector field, which measures the rotation or swirling strength of the field at a point.
• Understanding vector calculus concepts is pivotal in any field dealing with three-dimensional vectors, as it provides the mathematical foundation for describing changes and interactions within a field.

A deeper grasp of these operations allows for solving complex problems involving vectors and understanding how these apply to real-world scenarios, providing more insight into areas beyond just mathematics.

Three-Dimensional Vectors

Vectors in three-dimensional space have characteristics and properties that make them distinct from their lower-dimensional counterparts. They are used to represent physical quantities with both direction and magnitude, such as forces and velocities.

• A vector is denoted by its components along the x, y, and z axes, written as \( \langle x, y, z \rangle \) in 3D space.
• Operations like addition, scalar multiplication, and the dot and cross products are fundamental manipulations you can perform with vectors.
• The cross product's significance arises due to its perpendicularity to the original vectors, an attribute that enables visualization and computation of spatial orientations and directions.

Three-dimensional vectors are foundational in physics, mechanics, and engineering. Often, these applications require calculating moments, angles, and resultant forces, making understanding of vector operations, like the cross product, essential.