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Magazine Chapter 12: Problem 2
Find the cross product of \langle 1,0,2\rangle and \(\langle-1,-2,4\rangle .\)
Short Answer
Expert verified
The cross product is \( \langle 4, -6, -2 \rangle \).
Step by step solution
01
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 \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \).
02
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 \( b_1 = -1, b_2 = -2, b_3 = 4 \).
03
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: \( a_1b_2 - a_2b_1 = 1\cdot(-2) - 0\cdot(-1) = -2 + 0 = -2 \).
04
Write the Resulting Cross Product
Combine the calculated components into the resulting vector:\[ \mathbf{a} \times \mathbf{b} = \langle 4, -6, -2 \rangle \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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鈥攅ach 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.
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.
