Problem 9 - Precalculus: Functions and Graphs

Question

Show that the vectors are orthogonal. $$<4,-1>, \quad<2,8>$$

Step-by-Step Solution

Answer

The vectors are orthogonal because their dot product is zero.

Step 1: Understand the Definition of Orthogonal Vectors

Two vectors are orthogonal if their dot product is zero. We need to calculate the dot product of the given vectors to determine if they are orthogonal.

Step 2: Write the Formula for the Dot Product

The formula for the dot product of two vectors \( \langle a_1, a_2 \rangle \) and \( \langle b_1, b_2 \rangle \) is given by:\[ a_1 \cdot b_1 + a_2 \cdot b_2 \]

Step 3: Substitute the Given Vectors into the Formula

Let's substitute the components of the vectors \( \langle 4, -1 \rangle \) and \( \langle 2, 8 \rangle \) into the dot product formula:\[ 4 \cdot 2 + (-1) \cdot 8 \]

Step 4: Calculate the Dot Product

Now, calculate the value of the dot product:\[ 4 \cdot 2 + (-1) \cdot 8 = 8 - 8 = 0 \]

Step 5: Conclusion about Orthogonality

Since the dot product is equal to zero, the vectors \( \langle 4, -1 \rangle \) and \( \langle 2, 8 \rangle \) are orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It serves as a measure of how two vectors align with each other. Calculating the dot product involves multiplying corresponding components of the vectors and then summing the results. For two-dimensional vectors \( \langle a_1, a_2 \rangle \) and \( \langle b_1, b_2 \rangle \), the dot product is computed as:\[ a_1 \cdot b_1 + a_2 \cdot b_2 \]This operation is particularly useful to determine if two vectors are perpendicular or orthogonal. When the dot product equals zero, it means the vectors have no projection onto each other and they are orthogonal.

• Orthogonal vectors signify a 90-degree angle between them.
• They can exist in any number of dimensions.

Understanding the calculation and implications of the dot product is essential in both precalculus and other advanced mathematical applications.

Vectors in Precalculus

In precalculus, vectors are primarily used to represent quantities that have both magnitude and direction. Examples include velocity, force, and displacement. A vector can be expressed in component form, such as \( \langle x, y \rangle \), and is visually represented by an arrow.

• The direction of a vector is where the arrow points.
• The magnitude, or length, depends on its components and can be calculated using the Pythagorean theorem: \( \sqrt{x^2 + y^2} \).

This foundational concept prepares students for calculus and physics, where vectors become crucial in understanding motion and forces. In the context of this exercise, vectors demonstrate how different components interact when combined or analyzed. The concept of the dot product and orthogonality further enhances the student's toolkit in solving real-world problems.

Mathematical Proof

Mathematical proofs play a crucial role in validating mathematical statements, ensuring the results are reliable and logical. In the given exercise, proving that vectors are orthogonal involves logical reasoning using algebraic manipulation.First, remember the definition: vectors are orthogonal if their dot product is zero. Hence, proving orthogonality, in this case, reduces to computing the dot product of given vectors.
By evaluating:\[ 4 \cdot 2 + (-1) \cdot 8 = 8 - 8 = 0 \]we see that the result equals zero, confirming that the vectors are orthogonal.

• Each step in a mathematical proof follows logically from the previous ones.
• Proofs ensure that results are consistent and universally applicable.
• They form a foundation upon which more complex mathematical concepts are built.

Using such structured methodologies assists students in developing critical thinking and problem-solving skills, which are vital throughout their mathematical education.