Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the vectors \(\mathbf{a}=\langle 1,1,1\rangle, \mathbf{b}=\langle 1,-1,0\rangle\), and \(\mathbf{c}=\langle-1,-1,2\rangle\) are mutually orthogonal, that is, each pair of vectors is orthogonal.

Step-by-Step Solution

Answer

All three pairs of vectors are orthogonal as their dot products are zero.

Step 1: Define Orthogonality

Two vectors are orthogonal if their dot product is zero. We need to check the dot product of three pairs of vectors: \( \mathbf{a} \) with \( \mathbf{b} \), \( \mathbf{a} \) with \( \mathbf{c} \), and \( \mathbf{b} \) with \( \mathbf{c} \).

Step 2: Calculate Dot Product of \( \mathbf{a} \) and \( \mathbf{b} \)

The dot product is calculated as follows: \( \mathbf{a} \cdot \mathbf{b} = (1)(1) + (1)(-1) + (1)(0) \). Simplifying, we get: \( 1 - 1 + 0 = 0 \). The result is zero, so \( \mathbf{a} \) is orthogonal to \( \mathbf{b} \).

Step 3: Calculate Dot Product of \( \mathbf{a} \) and \( \mathbf{c} \)

The dot product is calculated as follows: \( \mathbf{a} \cdot \mathbf{c} = (1)(-1) + (1)(-1) + (1)(2) \). Simplifying, we get: \( -1 - 1 + 2 = 0 \). The result is zero, so \( \mathbf{a} \) is orthogonal to \( \mathbf{c} \).

Step 4: Calculate Dot Product of \( \mathbf{b} \) and \( \mathbf{c} \)

The dot product is calculated as follows: \( \mathbf{b} \cdot \mathbf{c} = (1)(-1) + (-1)(-1) + (0)(2) \). Simplifying, we get: \( -1 + 1 + 0 = 0 \). The result is zero, so \( \mathbf{b} \) is orthogonal to \( \mathbf{c} \).

Key Concepts

Dot Product

The dot product is a crucial concept in vector algebra that helps determine how two vectors relate to each other. It is also known as the scalar product, because it produces a single number from two vectors. To calculate the dot product of two vectors, each of their corresponding components is multiplied and then summed up. For example, if we have two vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\), their dot product is:\[ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \]If the result of the dot product is zero, the vectors are orthogonal, meaning they are perpendicular to each other at a right angle.

• Direction Relation: The dot product helps to understand the angle between two vectors.
• Magnitude Impact: If one of the vectors is scaled, the dot product scales by the same factor.

Vector Algebra

Vector algebra consists of various operations that can be performed on vectors, such as addition, subtraction, and multiplication.These operations are fundamental tools for analyzing vectors in mathematics and physics. A vector is a mathematical entity that has both a magnitude and direction, often represented in components along axes in a coordinate system.

Basic Operations

• Addition: Vectors can be added together by summing their corresponding components. For instance, \(\mathbf{a} = \langle 1, 2, 3 \rangle\) and \(\mathbf{b} = \langle 4, 5, 6 \rangle\), then \(\mathbf{a} + \mathbf{b} = \langle 1+4, 2+5, 3+6 \rangle = \langle 5, 7, 9 \rangle\).
• Subtraction: Similar to addition, subtracting vectors involves subtracting their corresponding components.
• Scalar Multiplication: Multiplying a vector by a scalar (a real number) scales each component of the vector by that scalar.

Understanding these operations allows for complex computations and analyses in physics, engineering, and computer science.

Mutual Orthogonality

Mutual orthogonality refers to a set of vectors being orthogonal to each other. This means any pair of vectors from the set will have a dot product of zero. In practice, checking mutual orthogonality involves computing the dot product for every possible pair in the vector set.

Significance in Mathematics and Applications

• Coordinate Systems: Orthogonal vectors are essential in defining Cartesian coordinate systems, where axes are perpendicular to each other.
• Signal Processing and Data Science: In these fields, orthogonal vectors are used to isolate independent components, simplifying analysis.
• Linear Algebra: In linear algebra, a set of mutually orthogonal vectors forms an orthogonal basis, which greatly simplifies matrix operations.

Understanding mutual orthogonality helps in maintaining independence among vectors, aiding in optimizing various computational techniques.