Problem 53 - Trigonometry

Question

Which pair of vectors are perpendicular? a. \(3 \mathbf{i}+4 \mathbf{j}\) and \(8 \mathbf{i}-6 \mathbf{j}\) b. \(3 \mathbf{i}+2 \mathbf{j}\) and \(2 \mathbf{i}+3 \mathbf{j}\) c. \(\mathbf{i}+5 \mathbf{j}\) and \(\mathbf{i}-5 \mathbf{j}\) d. \(2 \mathbf{i}-5 \mathbf{j}\) and \(-7 \mathbf{i}-3 \mathbf{j}\)

Step-by-Step Solution

Answer

Pair a is perpendicular, with a dot product of zero.

Step 1: Understand Perpendicular Vectors

Two vectors are perpendicular if their dot product is zero. The dot product of vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) is given by \( a_1b_1 + a_2b_2 \). Our task is to find the pair (if any) for which this calculation equals zero.

Step 2: Calculate Dot Product for Pair a

Consider vectors \(3 \mathbf{i} + 4 \mathbf{j}\) and \(8 \mathbf{i} - 6 \mathbf{j}\). Here, \(a_1 = 3\), \(a_2 = 4\), \(b_1 = 8\), and \(b_2 = -6\). The dot product is: \[3 \times 8 + 4 \times (-6) = 24 - 24 = 0\]So, these vectors are perpendicular.

Step 3: Calculate Dot Product for Pair b

Consider vectors \(3 \mathbf{i} + 2 \mathbf{j}\) and \(2 \mathbf{i} + 3 \mathbf{j}\). Here, \(a_1 = 3\), \(a_2 = 2\), \(b_1 = 2\), and \(b_2 = 3\). The dot product is: \[3 \times 2 + 2 \times 3 = 6 + 6 = 12\]These vectors are not perpendicular.

Step 4: Calculate Dot Product for Pair c

Consider vectors \(\mathbf{i} + 5 \mathbf{j}\) and \(\mathbf{i} - 5 \mathbf{j}\). Here, \(a_1 = 1\), \(a_2 = 5\), \(b_1 = 1\), and \(b_2 = -5\). The dot product is: \[1 \times 1 + 5 \times (-5) = 1 - 25 = -24\]These vectors are not perpendicular.

Step 5: Calculate Dot Product for Pair d

Consider vectors \(2 \mathbf{i} - 5 \mathbf{j}\) and \(-7 \mathbf{i} - 3 \mathbf{j}\). Here, \(a_1 = 2\), \(a_2 = -5\), \(b_1 = -7\), and \(b_2 = -3\). The dot product is: \[2 \times (-7) + (-5) \times (-3) = -14 + 15 = 1\]These vectors are not perpendicular.

Key Concepts

Dot Product

The dot product is a mathematical operation that takes two vectors and returns a single number. This number can tell us useful information about the relationship between the two vectors. In terms of geometry, the dot product helps us determine if the vectors are perpendicular, meaning they meet at a right angle. To find the dot product, you multiply the corresponding components of the vectors and then add those products together.

Consider two vectors:

• \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \)
• \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \)

The dot product is calculated as \( a_1b_1 + a_2b_2 \).

A very important property of the dot product is that if it equals zero, the vectors are perpendicular. For example, vectors \( 3\mathbf{i} + 4\mathbf{j} \) and \( 8\mathbf{i} - 6\mathbf{j} \) have a dot product of \( 3 \times 8 + 4 \times (-6) = 0 \). Thus, they are perpendicular.

Vector Arithmetic

Vector arithmetic involves operations like addition, subtraction, and scalar multiplication. These operations are straightforward yet foundational for understanding more complex concepts.

• Addition: Adding two vectors involves adding their corresponding components. For vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the result is \( (a_1 + b_1) \mathbf{i} + (a_2 + b_2) \mathbf{j} \).
• Subtraction: Similar to addition, but subtract the components instead. \( \mathbf{a} - \mathbf{b} \) results in \( (a_1 - b_1) \mathbf{i} + (a_2 - b_2) \mathbf{j} \).
• Scalar multiplication: Involves multiplying each component of the vector by a scalar (a constant number), \( k \mathbf{a} = (k \cdot a_1) \mathbf{i} + (k \cdot a_2) \mathbf{j} \).

Understanding these basic operations allows you to manipulate and combine vectors in various ways, paving the way to explore more complex tasks like resolving vector components or determining orthogonality (perpendicularity).

Vector Components

Vector components represent the projections of a vector along the coordinate axes. For a vector described in 2D space, like \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \), \( a_1 \) and \( a_2 \) are the respective components along the x-axis and y-axis.

Breaking a vector into its components is critical for various applications:

• Determining Length: The length (or magnitude) of vector \( \mathbf{a} \) is obtained by \( \sqrt{a_1^2 + a_2^2} \). This formula utilizes the Pythagorean theorem to integrate the components into a meaningful numerical value.
• Analyzing Direction: Components help illustrate the vector's direction by showing how much it extends in each axis’ direction.
• Facilitating Calculations: Knowing components simplifies operations like the dot product calculation since components directly correspond to the terms involved.

Understanding each vector component’s implication allows insight into the vector's effect and contribution to a system or equation.