Chapter 6: Problem 1

Which of the following pairs of vectors is orthogonal: 1\. \(x=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right], y=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]\) 2\. \(x=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right], y=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]\) 3\. \(x=e_{1}, y=e_{3}\) 4\. \(x=\left[\begin{array}{l}a \\ b\end{array}\right], y=\left[\begin{array}{c}-b \\ a\end{array}\right]\)

Short Answer

Expert verified

Pairs 1, 3, and 4 are orthogonal.

Step by step solution

- Understand Orthogonality

Two vectors are orthogonal if their dot product is zero. For vectors \( x \) and \( y \), this is expressed mathematically as \( x \boldsymbol{\bullet} y = 0 \).

- Calculate Dot Product for Choice 1

For vectors \( x = \left[ \begin{array}{l} 1 \ 0 \ 1 \end{array} \right] \) and \( y = \left[ \begin{array}{l} 0 \ 1 \ 0 \end{array} \right] \):\[ x \boldsymbol{\bullet} y = 1 \times 0 + 0 \times 1 + 1 \times 0 = 0 \]Since the result is zero, they are orthogonal.

- Calculate Dot Product for Choice 2

For vectors \( x = \left[ \begin{array}{l} 1 \ 1 \ 0 \end{array} \right] \) and \( y = \left[ \begin{array}{l} 1 \ 1 \ 1 \end{array} \right] \):\[ x \boldsymbol{\bullet} y = 1 \times 1 + 1 \times 1 + 0 \times 1 = 1 + 1 + 0 = 2 \]Since the result is not zero, they are not orthogonal.

- Understand Standard Basis Vectors

The standard basis vectors \( e_i \) for \( i = 1, 2, 3 \) has the form where only the \( i \)-th component is 1 and the rest are 0. E.g., \( e_1 = \left[ \begin{array}{c} 1 \ 0 \ 0 \end{array} \right] \) and \( e_3 = \left[ \begin{array}{c} 0 \ 0 \ 1 \end{array} \right] \). For \( x = e_1 \) and \( y = e_3 \):

- Calculate Dot Product for Choice 3

For vectors \( x = e_1 \) and \( y = e_3 \):\[ x \boldsymbol{\bullet} y = 1 \times 0 + 0 \times 0 + 0 \times 1 = 0 \]Since the result is zero, they are orthogonal.

- Calculate Dot Product for Choice 4

For vectors \( x = \left[ \begin{array}{l} a \ b \end{array} \right] \) and \( y = \left[ \begin{array}{c} -b \ a \end{array} \right] \):\[ x \boldsymbol{\bullet} y = a \times (-b) + b \times a = -ab + ab = 0 \]Since the result is zero, they are orthogonal.

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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 crucial operation in linear algebra and vector calculus. It helps determine whether two vectors are orthogonal or not. For two vectors, say \(x\) and \(y\), the dot product is defined as:
\[ x \boldsymbol{\bullet} y = x_1 y_1 + x_2 y_2 + \frac{\text{...}}{\text{...}} + x_n y_n \]
where \(x_i\) and \(y_i\) are the components of vectors \(x\) and \(y\), respectively. If the dot product of two vectors is zero, this indicates that the vectors are orthogonal. Such an understanding simplifies many problems in vector analysis and is foundational in higher-level math.

linear algebra

Linear algebra is an area of mathematics concerned with vector spaces and linear mappings between these spaces. It includes vectors, matrices, and vector equations. Understanding orthogonality and dot products are fundamental in linear algebra. Orthogonal vectors, for instance, are vital in decomposing vector spaces and solving systems of linear equations. It allows us to work in a clearer and more efficient manner, especially when working with high-dimensional data.

vector calculus

Vector calculus is a branch of mathematics that deals with vector fields and differential operators. It extends the concepts covered in linear algebra into multidimensional space. Orthogonality and the dot product are crucial in vector calculus for finding projections and performing operations like gradient, divergence, and curl. For example, computing the dot product between gradient vectors helps in understanding directional derivatives and provides insights into the functions' behavior.

standard basis vectors

Standard basis vectors are a set of vectors that form the basis for vector space. Each standard basis vector has a value of 1 in one component and 0 in all others. For a 3-dimensional space, the standard basis vectors are:
\[ e_1 = \begin{bmatrix} 1 \ 0 \ 0 \ \text{...} \end{bmatrix}, \ e_2 = \begin{bmatrix} 0 \ 1 \ 0 \ \text{...} \end{bmatrix}, \ e_3 = \begin{bmatrix} 0 \ 0 \ 1 \ \text{...} \end{bmatrix} \]
These vectors are orthogonal to each other because their dot product is zero. Standard basis vectors form a convenient way to describe any vector in the space as a combination of these basis vectors.

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Short Answer

Step by step solution

- Understand Orthogonality

- Calculate Dot Product for Choice 1

- Calculate Dot Product for Choice 2

- Understand Standard Basis Vectors

- Calculate Dot Product for Choice 3

- Calculate Dot Product for Choice 4

Key Concepts

dot product

linear algebra

vector calculus

standard basis vectors

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