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Chapter 2: Problem 142

For the following exercises determine whether the given vectors are orthogonal. \(\mathbf{a}=\langle x, x\rangle, \quad \mathbf{b}=\langle- y, y\rangle,\) where \(x\) and \(y\) are nonzero real numbers

Short Answer

Expert verified
The vectors are orthogonal.

Step by step solution

01

Understanding Orthogonality

Two vectors are orthogonal if their dot product is zero. Given vectors \( \mathbf{a} = \langle x, x \rangle \) and \( \mathbf{b} = \langle -y, y \rangle \), we need to calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \) and check if it equals zero.
02

Calculate Dot Product

The dot product of two vectors \( \mathbf{a} = \langle x_1, x_2 \rangle \) and \( \mathbf{b} = \langle y_1, y_2 \rangle \) is calculated as \( x_1y_1 + x_2y_2 \). For \( \mathbf{a} = \langle x, x \rangle \) and \( \mathbf{b} = \langle -y, y \rangle \), the components are \( x_1 = x, x_2 = x, y_1 = -y, \) and \( y_2 = y \). Thus, the dot product is \( x(-y) + x(y) \).
03

Simplify the Dot Product

Simplify the expression: \( x(-y) + x(y) = -xy + xy = 0 \). The result of the dot product is 0, which indicates that the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal.
04

Conclusion on Orthogonality

Since the dot product \( \mathbf{a} \cdot \mathbf{b} = 0 \), the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal. This occurs if and only if the vectors are perpendicular to each other.

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Key Concepts

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

Dot Product
The dot product is a fundamental operation in vector calculus that combines two vectors to yield a scalar value. This operation provides insight into the relationship between two vectors, specifically their geometric alignment.
  • To compute the dot product of two vectors \(\mathbf{a} = \langle x_1, x_2 \rangle\) and \(\mathbf{b} = \langle y_1, y_2 \rangle\), you multiply their corresponding components and sum the results: \(x_1y_1 + x_2y_2\).
  • If the resulting scalar value is zero, it indicates that the vectors are orthogonal.
  • The dot product is often used in physics and engineering to determine angles between vectors.
Overall, the dot product is a powerful tool in determining orthogonality, as a zero result directly translates to the vectors being perpendicular.
Vector Calculus
Vector calculus is an area of mathematics focused on vector fields and differential operators applied to vectors. It extends basic vector algebra and is indispensable for solving physical problems across many domains.
  • Vector calculus provides tools to handle quantities that have both magnitude and direction, such as velocity and force.
  • Key operations include the dot product, cross product, gradient, divergence, and curl, each serving a different purpose in analyzing vector behaviors.
  • For calculation of orthogonality, vector calculus employs operations like the dot product to assess alignment.
Comprehending how different vector calculus operations work together enhances understanding of complex systems in physics and engineering.
Perpendicular Vectors
Perpendicular vectors are vectors that intersect at a right angle (90 degrees) forming what is known as an orthogonal relationship.
  • This relationship is crucial, as it simplifies many complex mathematical and physical problems.
  • For two vectors \( \mathbf{a} \) and \( \mathbf{b} \), orthogonality is confirmed when their dot product equals zero, \(\mathbf{a} \cdot \mathbf{b} = 0\).
  • This property allows for easier computation in fields like graphics where determining perpendicularity is common.
Understanding perpendicular vectors involves recognizing their significance in simplifying vector additions, modeling, and transformations.

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Most popular questions from this chapter

For the following exercises, convert the given equations from cylindrical or spherical coordinates to rectangular coordinates. Identify the given surface. $$\rho^{2}\left(\sin ^{2}(\varphi)-\cos ^{2}(\varphi)\right)=1$$

For the following exercises, the equation of a plane is given. a. Find normal vector \(\mathbf{n}\) to the plane. Express \(\mathbf{n}\) using standard unit vectors. b. Find the intersections of the plane with the axes of coordinates. c. Sketch the plane. \(x+z=0\)

Determine whether the line of parametric equations \(x=1+2 t, y=-2 t, z=2+t, \quad t \in \mathbb{R}\) intersects the plane with equation \(3 x+4 y+6 z-7=0\) . If it does intersect, find the point of intersection.

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample. The symmetric equation for the line of intersection between two planes \(x+y+z=2\) and \(x+2 y-4 z=5\) is given by \(-\frac{x-1}{6}=\frac{y-1}{5}=z\).

The force vector \(\mathbf{F}\) acting on a proton with an electric charge of \(1.6 \times 10^{-19} \mathrm{C}\) (in coulombs) moving in a magnetic field \(\mathbf{B}\) where the velocity vector \(\mathbf{v}\) is given by \(\mathbf{F}=1.6 \times 10^{-19}(\mathbf{v} \times \mathbf{B})\) (here, \(\mathbf{v}\) is expressed in meters per second, \(\mathbf{B}\) is in tesla \([\mathrm{T}],\) and \(\mathbf{F}\) is in newtons \([\mathrm{N}] )\) . Find the force that acts on a proton that moves in the \(x y\) -plane at velocity \(\mathbf{v}=10^{5} \mathbf{i}+10^{5} \mathbf{j}\) (in meters per second) in a magnetic field given by \(\mathbf{B}=0.3 \mathbf{j} .\)
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