Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9 -12: Show that the vectors are orthogonal. \(\langle 3,6\rangle\) \(\langle 4,-2\rangle\)

Step-by-Step Solution

Answer

The vectors are orthogonal because their dot product is 0.

Step 1: Recall the definition of orthogonality

Two vectors are orthogonal if their dot product is zero. The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is calculated as \( a \cdot c + b \cdot d \).

Step 2: Calculate the dot product

For the given vectors \( \langle 3, 6 \rangle \) and \( \langle 4, -2 \rangle \), compute the dot product: \[ 3 \times 4 + 6 \times (-2) \].

Step 3: Simplify the calculations

Perform the multiplication and addition: \[ 3 \times 4 = 12 \]\[ 6 \times (-2) = -12 \]Then add them together: \[ 12 + (-12) = 0 \].

Step 4: Conclude about orthogonality

Since the dot product is \( 0 \), the vectors \( \langle 3, 6 \rangle \) and \( \langle 4, -2 \rangle \) are orthogonal to each other.

Key Concepts

Understanding the Dot Product

The dot product is a fundamental operation in vector arithmetic that helps us determine if two vectors are orthogonal. This operation takes two vectors and returns a single number. Here’s how it works:

• For two vectors, say \( \langle a, b \rangle \) and \( \langle c, d \rangle \), their dot product is calculated as \( a \cdot c + b \cdot d \).
• The dot product is a measure of how much one vector extends in the direction of another.

The key property of the dot product is its ability to indicate orthogonality. If the vectors are orthogonal, their dot product will be zero. This is a very practical way to confirm orthogonality since it simplifies the process to a basic arithmetic calculation. In our exercise, the dot product calculation was \( 12 + (-12) = 0 \), confirming that the vectors are orthogonal.

Basics of Vector Arithmetic

Vector arithmetic involves operations like addition, subtraction, and scalar multiplication. These operations are fundamental in understanding vectors and their interactions. Let's break down these operations:

• Addition: To add vectors, you simply add the corresponding components. For instance, \( \langle a, b \rangle + \langle c, d \rangle = \langle a+c, b+d \rangle \).
• Subtraction: Similarly, subtracting vectors involves subtracting their corresponding components. It follows the formula \( \langle a, b \rangle - \langle c, d \rangle = \langle a-c, b-d \rangle \).
• Scalar multiplication: This involves multiplying each component of a vector by a scalar value, \( k \), thus \( k \cdot \langle a, b \rangle = \langle ka, kb \rangle \).

Applications of vector arithmetic are numerous, especially in fields like physics and engineering where vectors are used to represent quantities like force and velocity. Knowing how to handle these basic operations is crucial for manipulating vectors and understanding their properties.

Algebra in Vector Calculations

Algebra plays a significant role in vector calculations, helping simplify processes and solve for unknowns. When dealing with vectors, we often rely heavily on algebraic principles. Here are a few ways algebra integrates into vector calculations:

• Using algebra to solve equations involving vectors, such as finding unknown vector components given some conditions.
• Simplifying expressions derived from vector operations, like calculating the dot product.
• Understanding linear combinations of vectors, which assists in defining concepts such as vector spaces and vector projections.

In our example, algebra helped simplify the dot product operation. We computed \( 3 \times 4 + 6 \times (-2) \), which was straightforward thanks to algebraic multiplication and addition. Mastering these skills helps in making vector-related computations easier and more intuitive, paving the way for tackling more complex mathematical problems involving vectors.