Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 17-20: Determine \(m\) such that the two vectors are orthogonal. $$ 9 \mathbf{i}-16 m \mathbf{j}, \quad \mathbf{i}+4 m \mathbf{j} $$

Step-by-Step Solution

Answer

\( m = \pm \frac{3}{8} \).

Step 1: Understand the condition for orthogonality

Two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \) are orthogonal if their dot product is zero: \( a_1b_1 + a_2b_2 = 0 \).

Step 2: Identify components of the vectors

The vectors given are \( 9 \mathbf{i} - 16m \mathbf{j} \) and \( \mathbf{i} + 4m \mathbf{j} \). Thus, the components are: \( a_1 = 9 \), \( a_2 = -16m \), \( b_1 = 1 \), and \( b_2 = 4m \).

Step 3: Write the dot product equation

The dot product of the vectors is: \( 9 \cdot 1 + (-16m) \cdot (4m) = 9 - 64m^2 \).

Step 4: Set the dot product to zero for orthogonality

For the vectors to be orthogonal, we set the equation from Step 3 to zero: \( 9 - 64m^2 = 0 \).

Step 5: Solve for \( m \)

To solve \( 9 - 64m^2 = 0 \), begin by solving for \( m^2 \): \( 64m^2 = 9 \). Divide both sides by 64 to get \( m^2 = \frac{9}{64} \). Taking the square root of both sides results in \( m = \pm \frac{3}{8} \).

Key Concepts

Understanding the Dot Product

The dot product is an important concept in vector mathematics. It helps us determine if two vectors are orthogonal, meaning they are at right angles to each other. For two 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 by multiplying the corresponding components and adding them together. This gives us the expression:

• \( a_1b_1 + a_2b_2 \)

If the result of the dot product is zero, the vectors are orthogonal. This is a quick method to check angles between vectors without having to draw them.

Exploring Vector Components

Vectors have both direction and magnitude. They are often broken down into components, making calculations easier. Each vector component aligns with a coordinate axis, often labeled as \( \mathbf{i} \) and \( \mathbf{j} \) for simplicity in a 2D plane. For instance, consider the vectors \( 9 \mathbf{i} - 16m \mathbf{j} \) and \( \mathbf{i} + 4m \mathbf{j} \). The individual components are:

• For the first vector:
  • \( a_1 = 9 \)
  • \( a_2 = -16m \)
• For the second vector:
  • \( b_1 = 1 \)
  • \( b_2 = 4m \)

Breaking down vectors into components helps when calculating dot products or resolving equations related to angles and magnitudes.

Solving Equations for Vector Problems

In vector problems where you need to determine if two vectors are orthogonal, you often end up with equations that you need to solve. Let's follow solving such an equation for the given vectors. Once we have the vector components, we use them to create an equation from their dot product. For the vectors \( 9 \mathbf{i} - 16m \mathbf{j} \) and \( \mathbf{i} + 4m \mathbf{j} \), the dot product equation is:

• \( 9 \cdot 1 + (-16m) \cdot (4m) = 9 - 64m^2 \)

To find \( m \) that makes the vectors orthogonal, set this equation to zero:

• \( 9 - 64m^2 = 0 \)

Now solve for \( m \):

• Start with \( 64m^2 = 9 \).
• Divide through by 64, giving \( m^2 = \frac{9}{64} \).
• Finally, taking the square root gives \( m = \pm \frac{3}{8} \).

Solving such equations is a matter of logical steps and ensuring you follow algebraic principles correctly. This approach ensures you find the required values efficiently.