Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the vectors \(\langle 6,3\rangle\) and \(\langle-1,2\rangle\) are orthogonal.

Step-by-Step Solution

Answer

The vectors are orthogonal because their dot product is zero.

Step 1: Understanding Orthogonal Vectors

Vectors are orthogonal if their dot product is equal to zero. If two vectors \( \textbf{a} = \langle a_1, a_2 \rangle \) and \( \textbf{b} = \langle b_1, b_2 \rangle \) are orthogonal, then it must be that \( a_1b_1 + a_2b_2 = 0 \).

Step 2: Setting Up the Dot Product

For the given vectors \( \langle 6, 3 \rangle \) and \( \langle -1, 2 \rangle \), we need to calculate their dot product. This is done by multiplying their corresponding components and then adding the products: \( 6 \times (-1) + 3 \times 2 \).

Step 3: Calculating the Dot Product

Compute the products of the corresponding components: - \( 6 \times (-1) = -6 \)- \( 3 \times 2 = 6 \)Now, add these results together: \( -6 + 6 = 0 \).

Step 4: Conclusion on Orthogonality

Since the dot product of \( \langle 6, 3 \rangle \) and \( \langle -1, 2 \rangle \) equals zero, the vectors are orthogonal according to the definition.

Key Concepts

Understanding the Dot Product

The dot product is a fundamental operation in vector algebra. It provides a way to multiply two vectors together to produce a scalar. The dot product for two-dimensional vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is calculated as:

• \( a_1 \times b_1 + a_2 \times b_2 \)

This formula multiplies the corresponding components of the vectors and then sums their products. The result is a single number or scalar.

A key insight into the dot product is its relation to the angle between two vectors. Specifically, if the dot product is zero, it implies that the vectors are orthogonal or perpendicular to each other.

For example, if we calculate the dot product of \( \langle 6, 3 \rangle \) and \( \langle -1, 2 \rangle \), we get:

• \( 6 \times (-1) + 3 \times 2 = -6 + 6 = 0 \).

Since it equals zero, these vectors are orthogonal.

An Introduction to Vector Algebra

Vector algebra is the branch of mathematics that deals with quantities that have both a magnitude and a direction. Vectors can represent a range of phenomena, from physics (like forces) to computer graphics.

To manipulate vectors, you need to understand various operations, such as addition, subtraction, and scalar multiplication. For instance, if we have two vectors \( \mathbf{u} \) and \( \mathbf{v} \), their sum is found by adding their corresponding components.

• Addition: \( \mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle \)
• Subtraction: \( \mathbf{u} - \mathbf{v} = \langle u_1 - v_1, u_2 - v_2 \rangle \)
• Scalar Multiplication: \( c \cdot \mathbf{u} = \langle c \times u_1, c \times u_2 \rangle \)

Vector algebra allows us to solve complex problems in multiple dimensions by using these operations. It is crucial in fields ranging from engineering to economics.

The Role of Calculus in Vector Analysis

Calculus extends the power of vector algebra by introducing techniques such as differentiation and integration to functions with multiple variables.

When working with vectors, one often needs to explore how they change over time or space. In these cases, calculus's role is invaluable. Differentiation can be used to find the rate of change, while integration might be used to find the area or volume described by vectors.

For example:

• Differentiation of vector functions can help find velocity and acceleration in physics.
• Integration of vector fields can be used to compute things like work done by a force field.

These calculus tools allow us to model and analyze real-world phenomena effectively. Consequently, combining calculus with vector algebra is essential for deeper insights into fields like electromagnetism and fluid dynamics.