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Chapter 11: Problem 49

Define the cross product of vectors \(\mathbf{u}\) and \(\mathbf{v}\).

Short Answer

Expert verified
The cross product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by \(\mathbf{u}\times\mathbf{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)\). This new vector is perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\), and its magnitude is equal to the area of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\). The direction of the vector is determined by the right-hand rule.

Step by step solution

01

Identify the vectors

Given two vectors \(\mathbf{u}\) and \(\mathbf{v}\), let \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\). These are the vectors we will be working with.
02

Apply the cross product formula

The cross product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by the formula \(\mathbf{u}\times\mathbf{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)\).
03

Interpret the result

The resulting vector \(\mathbf{u}\times\mathbf{v}\) is perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\), and its magnitude is equal to the area of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\). The direction of \(\mathbf{u}\times\mathbf{v}\) is determined by the right-hand rule.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vectors
Vectors are essential mathematical entities used to represent quantities that possess both magnitude and direction. A vector is typically denoted with an arrow symbol and can be expressed in coordinate form, such as \( \mathbf{u} = (u_1, u_2, u_3) \). This notation indicates that the vector has three components in a three-dimensional space, corresponding to its projections on the x, y, and z axes, respectively.
  • Magnitude: The magnitude of a vector, represented by \( ||\mathbf{u}|| \), is the length or size of the vector. It can be calculated using the formula \( ||\mathbf{u}|| = \sqrt{u_1^2 + u_2^2 + u_3^2} \).
  • Direction: The direction is where the vector points. It is often described using angles or unit vectors.
Working with vectors allows us to perform operations like addition, subtraction, and multiplication, including the dot and cross products, which are foundational in physics and engineering to describe forces, motion, and more.
Perpendicular Vectors
Perpendicular vectors are vectors that intersect at a 90-degree angle. In mathematical terms, two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular, or orthogonal, if their dot product is zero: \( \mathbf{u} \cdot \mathbf{v} = 0 \).
  • When two vectors are perpendicular, they form the axes of a coordinate system. Each vector is independent, contributing to different dimensions of the space.
  • In the context of the cross product, the resulting vector is always perpendicular to the initial vectors, \( \mathbf{u} \) and \( \mathbf{v} \).
This perpendicular relationship is crucial in three-dimensional geometry and physics, enabling vector fields and the study of spaces determined by multiple independent directions.
Right-Hand Rule
The right-hand rule is a simple method used to determine the direction of the cross product vector. When finding \( \mathbf{u} \times \mathbf{v} \), orient your right hand so that your fingers point in the direction of the first vector, \( \mathbf{u} \).
  • Fingers: Point towards the direction of the first vector, \( \mathbf{u} \).
  • Palm Rotation: Swing your palm, along with your fingers, towards the direction of the second vector, \( \mathbf{v} \).
  • Thumb: The direction your thumb points is the direction of the cross product vector \( \mathbf{u} \times \mathbf{v} \).
This tool is particularly handy in physics for electromagnetism and mechanical contexts, where knowing the spatial orientation of forces or moments is significant.
Parallelogram Area
The area of a parallelogram formed by two vectors can be calculated using their cross product. When you perform a cross product \( \mathbf{u} \times \mathbf{v} \), the magnitude of the result is equal to the area of the parallelogram that spans the vectors.
  • The area is given by \( ||\mathbf{u} \times \mathbf{v}|| \), which represents the magnitude of the cross product vector.
  • This calculation reflects how much space the vectors cover when translated to form a parallelogram.
  • If the vectors are parallel, the area (and therefore the cross product) is zero, as they can't form a parallelogram.
Understanding the geometrical interpretation of cross products in terms of areas is not only important for geometry but also for solving practical problems in physics and engineering, where areas between forces and space coverage need to be determined.

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Most popular questions from this chapter

Find the point(s) of intersection (if any) of the plane and the line. Also determine whether the line lies in the plane.\(5 x+3 y=17, \quad \frac{x-4}{2}=\frac{y+1}{-3}=\frac{z+2}{5}\)

(a) find the unit tangent vectors to each curve at their points of intersection and (b) find the angles \(\left(0 \leq \theta \leq 90^{\circ}\right)\) between the curves at their points of intersection. $$ (y+1)^{2}=x, \quad y=x^{3}-1 $$

Find the distance between the point and the line given by the set of parametric equations.\((-2,1,3) ; \quad x=1-t, \quad y=2+t, \quad z=-2 t\)

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Distance Two insects are crawling along different lines in three-space. At time \(t\) (in minutes), the first insect is at the point \((x, y, z)\) on the line \(x=6+t, \quad y=8-t, \quad z=3+t\) Also, at time \(t\), the second insect is at the point \((x, y, z)\) on the line \(x=1+t, \quad y=2+t, \quad z=2 t\) Assume distances are given in inches. (a) Find the distance between the two insects at time \(t=0\). (b) Use a graphing utility to graph the distance between the insects from \(t=0\) to \(t=10\). (c) Using the graph from part (b), what can you conclude about the distance between the insects? (d) How close do the insects get?
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