Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9 -12: Show that the vectors are orthogonal. $$ 8 \mathbf{i}-4 \mathbf{j}, \quad-6 \mathbf{i}-12 \mathbf{j} $$

Step-by-Step Solution

Answer

The vectors are orthogonal because their dot product is zero.

Step 1: Understand the concept of orthogonality

Two vectors are orthogonal if their dot product is zero. We need to calculate the dot product of the given vectors \(8\mathbf{i} - 4\mathbf{j}\) and \(-6\mathbf{i} - 12\mathbf{j}\).

Step 2: Identify components of the vectors

The first vector is \(\mathbf{a} = 8\mathbf{i} - 4\mathbf{j}\), so its components are \(a_1 = 8\) and \(a_2 = -4\). The second vector is \(\mathbf{b} = -6\mathbf{i} - 12\mathbf{j}\), so its components are \(b_1 = -6\) and \(b_2 = -12\).

Step 3: Calculate the dot product

The dot product of two vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}\) is calculated as: \(a_1b_1 + a_2b_2\). Substituting the components, we have: \(8(-6) + (-4)(-12)\).

Step 4: Simplifying the dot product

Simplify the expression calculated in the previous step: \(8 \times -6 = -48\) and \(-4 \times -12 = 48\). Then add them: \(-48 + 48 = 0\).

Step 5: Conclusion

Since the dot product of the vectors is zero, the vectors \(8\mathbf{i} - 4\mathbf{j}\) and \(-6\mathbf{i} - 12\mathbf{j}\) are orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector algebra. It represents an operation that takes two vectors and returns a scalar. To compute the dot product, you multiply corresponding components of the vectors and sum the results. In mathematical terms, for vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the dot product is calculated as:

• \( a_1 b_1 + a_2 b_2 \)

This operation plays a crucial role in determining the angle between vectors. A key point is that when the dot product is zero, the vectors are orthogonal, meaning they are perpendicular to each other. This makes the dot product instrumental in identifying orthogonality in vector studies.

Vector Components

Vector components refer to the individual values that define a vector in a given coordinate system. For 2D vectors, these are usually represented along the x and y axes, often marked by \( \mathbf{i} \) and \( \mathbf{j} \) respectively. For example, in the vector \( 8 \mathbf{i} - 4 \mathbf{j} \), 8 is the component along the x-axis, and -4 is the component along the y-axis.

Understanding vector components is essential because they form the basis for operations like the dot product. By breaking down vectors into their components, calculations become straightforward. This method allows you to analyze their behavior and find relationships such as parallelism and orthogonality.

Orthogonality

Orthogonality is an important concept in vector algebra. Two vectors are defined as orthogonal if their dot product equals zero. This implies that they meet at a 90-degree angle, or in simpler terms, they are perpendicular. The idea is quite similar to the concept of perpendicular lines.

• If vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal, then \( a_1 b_1 + a_2 b_2 = 0 \).

This property is very useful in physics and engineering, as orthogonal vectors often represent independent directions. Remember, just checking the dot product is enough to confirm orthogonality; you don't need to draw the vectors.

Vector Algebra

Vector algebra is the field of mathematics that deals with vector operations and the vector space itself. It includes operations such as addition, subtraction, and dot product. Each of these operations is crucial for various applications in physics, geometry, and engineering.

Understanding vector algebra helps in solving complex problems by breaking them down into simpler operations using vectors and their properties. For example, using vector algebra, you can determine if vectors are orthogonal based on their components' interactions through dot products. This makes vector algebra a powerful tool in both theoretical and applied mathematics.