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Chapter 5: Problem 55

Determine all vectors \(v\) that are orthogonal to \(\mathbf{u}\). $$\mathbf{u}=(0,5)$$

Short Answer

Expert verified
The set of all vectors \(v = (x, 0)\), for all \(x\) in the real numbers, are orthogonal to \(\mathbf{u}\).

Step by step solution

01

Define \(\mathbf{u}\) and \(\mathbf{v}\)

In the given exercise, vector \(\mathbf{u}\) is defined as \((0, 5)\). We will define a vector \(v\) as \((x,y)\).
02

Define the dot product

The dot product of two vectors in 2D is given by \(\mathbf{u} \cdot \mathbf{v} = u_{1}v_{1} + u_{2}v_{2}\). In this case, it means \((0, 5) \cdot (x, y) = 0*x + 5*y\).
03

Calculate the dot product

Given that the dot product of orthogonal vectors is zero, we solve the equation for y in terms of x: 0*x + 5*y = 0. This leads to \(y = 0\). Therefore, any vector \(v\) of the form \((x, 0)\), where \(x\) can be any real number, will be orthogonal to the vector \(\mathbf{u}\).

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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 concept in vector operations. It provides a way to measure how much one vector extends in the direction of another vector. You calculate the dot product by multiplying corresponding components of two vectors and then summing those products.
  • For example, if we have two vectors in 2D, \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \), the dot product \( \mathbf{u} \cdot \mathbf{v} \) is calculated as: \( u_1 \cdot v_1 + u_2 \cdot v_2 \).
  • In the case of orthogonal vectors, their dot product is zero because they are perpendicular to each other.
  • Orthogonality implies no overlap in direction, hence no "work" done in the direction of the other vector.
To determine if two vectors are orthogonal, simply calculate their dot product. If it's zero, the vectors are orthogonal.
Linear Algebra
Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between these spaces. It is crucial in many fields of science and engineering.
  • Vectors and matrices are the main objects of study in linear algebra.
  • It includes operations such as addition and multiplication of matrices and vectors.
  • Vector spaces provide a framework for modeling multiple linear operations and transformations.
In the context of vectors, understanding linear algebra helps clarify concepts like the dot product, vector magnitude, and orthogonality, which are essential in understanding and performing vector operations. Moreover, solving equations involving vectors often requires applying linear algebra principles, such as in finding all orthogonal vectors to a given vector, as in our exercise.
Vector Operations
Vector operations include addition, subtraction, multiplication (including scalar and dot products), and finding magnitudes. These operations are essential tools in physics, engineering, and computer graphics.
  • Adding vectors involves adding their corresponding components.
  • Scalar multiplication is multiplying a vector by a number, scaling its magnitude without changing its direction.
  • The dot product, as discussed, involves multiplying corresponding components and summing them up.
  • Another key vector operation is finding orthogonal vectors. Vectors are orthogonal if their dot product is zero.
Understanding these operations allows us to solve a variety of problems involving vectors. For instance, in the exercise given, the goal is to find a vector orthogonal to \(\mathbf{u}\). By performing these vector operations, one can easily determine if vectors are orthogonal and solve related problems.

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Most popular questions from this chapter

Find bases for the four fundamental subspaces of the matrix \(A\). $$ A=\left[\begin{array}{rrr} 0 & -1 & 1 \\ 1 & 2 & 0 \\ 1 & 1 & 1 \end{array}\right] $$

Prove Lagrange's Identity: \(\mathbf{u} \times \mathbf{v}\left\|^{2}=\right\| \mathbf{u}\left\|^{2}\right\| \mathbf{v} \|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2}\)

Determine whether the vectors are orthogonal, parallel, or neither. Explain. $$\mathbf{u}=(-\sin \theta, \cos \theta, 1), \quad \mathbf{v}=(\sin \theta,-\cos \theta, 0)$$

Find the orthogonal projection of \(f\) onto \(g .\) Use the inner product in \(C[a, b]\) $$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x$$ $$C[-1,1], \quad f(x)=x, \quad g(x)=1$$

Find bases for the four fundamental subspaces of the matrix A listed below. \(N(A)=\) nullspace of \(A \quad N\left(A^{T}\right)=\) nullspace of \(A^{T}\) \(\boldsymbol{R}(\boldsymbol{A})=\) column space of \(\boldsymbol{A} \quad \boldsymbol{R}\left(\boldsymbol{A}^{T}\right)=\) column space of \(\boldsymbol{A}^{T}\) Then show that \(N(A)=R\left(A^{T}\right)^{\perp}\) and \(N\left(A^{T}\right)=R(A)^{\perp}\). \(\left[\begin{array}{rrr}1 & 1 & -1 \\ 0 & 2 & 1 \\ 1 & 3 & 0\end{array}\right]\)
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