Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Determine whether each pair of vectors is orthogonal. $$\langle 1,2\rangle,\langle- 6,3\rangle$$

Step-by-Step Solution

Answer

The vectors are orthogonal.

Step 1: Understanding Orthogonal Vectors

Two vectors are orthogonal if their dot product is zero. The dot product of vectors \( \langle a_1, a_2 \rangle \) and \( \langle b_1, b_2 \rangle \) is given by \( a_1 \times b_1 + a_2 \times b_2 \).

Step 2: Calculate the Dot Product

Calculate the dot product for the vectors \( \langle 1, 2 \rangle \) and \( \langle -6, 3 \rangle \). The dot product is calculated as: \( 1 \times (-6) + 2 \times 3 = -6 + 6 = 0 \).

Step 3: Conclusion about Orthogonality

Since the dot product of the vectors is zero, we conclude that the vectors \( \langle 1, 2 \rangle \) and \( \langle -6, 3 \rangle \) are orthogonal.

Key Concepts

Orthogonal Vectors

Understanding orthogonal vectors is quite simple and yet crucial in vector analysis. Two vectors are said to be orthogonal if they meet at a 90-degree angle. In geometry, this means the vectors are perpendicular to each other. The key property that helps determine orthogonality in algebraic terms is the dot product. When two vectors are orthogonal, their dot product equals zero. This condition makes calculations very straightforward. For instance, with vectors \( \langle 1, 2 \rangle \) and \( \langle -6, 3 \rangle \), the dot product is computed and found to be zero, confirming the vectors are orthogonal. So, whenever you need to check whether vectors are orthogonal, calculate their dot product: if it’s zero, they are indeed orthogonal.

Dot Product

The dot product, also known as the scalar product, is an essential operation in vector mathematics. It involves multiplying corresponding components of two vectors and then summing those products. For vectors \( \langle a_1, a_2 \rangle \) and \( \langle b_1, b_2 \rangle \), the dot product is calculated as \( a_1 \times b_1 + a_2 \times b_2 \). This results in a single number, or scalar.

The dot product has several applications:

• It helps in finding the angle between two vectors.
• It enables checking the orthogonality of vectors.
• It assists in projecting one vector onto another.

Using the example given in the exercise, \( \langle 1, 2 \rangle \) and \( \langle -6, 3 \rangle \), we compute the dot product as \( 1 \times (-6) + 2 \times 3 = -6 + 6 = 0 \), which plays a critical role in concluding the vectors are orthogonal.

Vector Components

Understanding vector components is vital for manipulating vectors effectively. A vector, generally represented as \( \langle a_1, a_2, ..., a_n \rangle \), consists of components along various axes. In two-dimensional space, vectors typically have two components—each representing the vector's projection on the x-axis and y-axis.

These components allow one to perform calculations such as the dot product or the magnitude of the vector. Breaking down a vector into these components provides a clearer picture of its direction and size in a coordinate system. For example, for the vector \( \langle 1, 2 \rangle \), \( 1 \) is the x-component and \( 2 \) is the y-component. This breakdown is important because each component carries part of the vector’s information, integral for many vector operations.