Problem 32 - Precalculus Mathematics for Calculus

Question

Determine whether or not the given vectors are perpendicular. $$\langle x,-2 x, 3 x\rangle,\langle 5,7,3)$$

Step-by-Step Solution

Answer

The vectors are perpendicular.

Step 1: Understand the Definition of Perpendicular Vectors

Two vectors are perpendicular if their dot product equals zero. The dot product of vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is calculated as \( a_1b_1 + a_2b_2 + a_3b_3 \).

Step 2: Identify the Components of the Vectors

Identify the components of the vectors given:- \( \mathbf{u} = \langle x, -2x, 3x \rangle \)- \( \mathbf{v} = \langle 5, 7, 3 \rangle \)

Step 3: Compute the Dot Product

Calculate the dot product for \( \mathbf{u} \) and \( \mathbf{v} \):\[ x\times 5 + (-2x)\times 7 + 3x\times 3 \]

Step 4: Simplify the Expression

Simplify the dot product expression:\( 5x - 14x + 9x = (5 - 14 + 9)x = 0x \).This simplifies to \( 0 \), indicating that the vectors are perpendicular.

Key Concepts

dot_product

The dot product is a fundamental operation in vector mathematics, expressing a specific kind of multiplication between two vectors. It yields a scalar (a single number) as the result, rather than another vector. This scalar gives us helpful information about the relationship between the vectors, particularly in determining if vectors are perpendicular.

The formula for computing the dot product of two 3D vectors, given vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), is:

• \( a_1b_1 + a_2b_2 + a_3b_3 \)

An important property of the dot product is its connection with angles. When the dot product of two vectors is zero, these vectors are orthogonal, meaning they meet at a right angle or, simply put, they are perpendicular. This characteristic is especially useful in geometry, physics, and various branches of engineering.

Thus, to ascertain the perpendicularity of vectors, like the ones in our exercise, we compute the dot product using their components, and check whether the result is zero.

vector_components

Understanding vector components is crucial for working with vectors, especially when computing things like the dot product. A vector component refers to each individual part of the vector that, when combined with the other parts, fully describes the vector in space.

For example, the vector \( \mathbf{u} = \langle x, -2x, 3x \rangle \) can be broken down as follows:

• First component: \( x \)
• Second component: \( -2x \)
• Third component: \( 3x \)

Similarly, the vector \( \mathbf{v} = \langle 5, 7, 3 \rangle \) has the components:

• First component: \( 5 \)
• Second component: \( 7 \)
• Third component: \( 3 \)

By identifying each component, we prepare to effectively calculate the dot product or any other vector operation required. Component analysis is fundamental in dividing vector problems into manageable parts, allowing us to handle each part independently before combining them back into the full solution.

calculating_vectors

Calculating vectors involves various mathematical operations that help us analyze and understand vector behaviors and properties. In our exercise, we aim to determine if two vectors are perpendicular through vector calculations like the dot product.

First, identify the components of each vector involved, as previously described. Then, apply the formula for the dot product:

• \( \mathbf{u} = \langle x, -2x, 3x \rangle \)
• \( \mathbf{v} = \langle 5, 7, 3 \rangle \)

For the dot product calculation, multiply corresponding components and sum the results. The calculation here is:

• \( 5x + (-2x) \times 7 + 3x \times 3 \)

This simplifies to:

• \( 5x - 14x + 9x \)
• Which results in: \( (5 - 14 + 9)x = 0x \)

The outcome is zero, confirming perpendicularity since the dot product equaling zero signals perpendicular vectors. This method, with correct component use, yields insights into vector orientation and interrelationships with ease.