Problem 26 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\langle 5,-2\rangle \text { and }\langle-5,2\rangle$$

Step-by-Step Solution

Answer

The vectors are not orthogonal because their dot product is -29.

Step 1: Understand Orthogonality

Two vectors are considered orthogonal if their dot product is zero. This means that they are perpendicular to each other in the geometric sense.

Step 2: Recall the Dot Product Formula

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 3: Substitute into the Dot Product Formula

Substitute the given vectors into the dot product formula: \[ \langle 5, -2 \rangle \text{ and } \langle -5, 2 \rangle \Rightarrow (5)(-5) + (-2)(2) \].

Step 4: Calculate the Dot Product

Perform the multiplication and addition:\[ 5 \times -5 = -25 \]\[ -2 \times 2 = -4 \]Add the results: \[ -25 + (-4) = -29 \].

Step 5: Determine Orthogonality

Since the dot product \(-29\) is not zero, the vectors \( \langle 5, -2 \rangle \) and \( \langle -5, 2 \rangle \) are not orthogonal.

Key Concepts

Dot Product

The dot product is a crucial concept in vector algebra, often used to determine relationships between vectors. It's a method of multiplying two vectors, resulting in a scalar value. The dot product can be calculated using the formula: \( a \cdot c + b \cdot d \), for two-dimensional vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \).
Here's a simple way to understand this:

• Take the corresponding components of each vector.
• Multiply them together.
• Add the results.

If the dot product is zero, the vectors are orthogonal, meaning they form a 90-degree angle and are perpendicular.
For example, in the exercise given, the vectors \( \langle 5, -2 \rangle \) and \( \langle -5, 2 \rangle \) had their components multiplied: \( 5 \times -5 = -25 \) and \( -2 \times 2 = -4 \). Adding \( -25 + (-4) \) equals \( -29 \), which shows the vectors are not orthogonal.

Vector Calculation

Vector calculations are fundamental in physics and engineering, involving operations like addition, subtraction, and finding the dot product. Vectors represent quantities with both magnitude and direction, such as force or velocity.
To perform vector calculations effectively, especially for the dot product, follow these steps:

• Identify the components of each vector; these are usually in the form \( \langle a, b \rangle \) for two-dimensional vectors.
• Apply operations like addition or dot product, based on the question's requirements.
• Make mental checks for each operation to avoid calculation errors.

Understanding how to manage these calculations prepares you not only to find solutions efficiently but also to comprehend their significance in multiple applications. In the given solution, by substituting and calculating the components correctly, we reached the conclusion about the orthogonality.

Perpendicular Vectors

Perpendicular vectors, or orthogonal vectors, are vectors that intersect at a right angle. The presence of a 90-degree angle between them is the defining characteristic, and it's verified through the dot product. When the dot product results in zero, this confirms the perpendicular nature.
To visualize this, consider vectors on a coordinate plane intersecting such that their combined angle measures 90 degrees.
These vectors have immense implications.

• They are crucial in physics where forces move in perpendicular directions.
• In computer graphics, they help in calculating lighting and shading by defining edges and surfaces.
• Engineers use them in mechanics to resolve forces and analyze equilibrium.

Thus, determining whether two vectors such as \( \langle 5, -2 \rangle \) and \( \langle -5, 2 \rangle \) are perpendicular requires the calculation of their dot product. Since the result \(-29\) is not zero, these vectors, as demonstrated, are not perpendicular.