Problem 34 - Trigonometry

Question

Show that each pair of vectors is perpendicular. \(-4 \mathbf{i}-3 \mathbf{j}\) and \(6 \mathbf{i}-8 \mathbf{j}\)

Step-by-Step Solution

Answer

The vectors are perpendicular because their dot product is 0.

Step 1: Identify the Vectors

The first vector is given as \(-4 \mathbf{i}-3 \mathbf{j}\) and can be represented in component form as \( \vec{a} = (-4, -3) \). The second vector is \(6 \mathbf{i}-8 \mathbf{j}\), represented as \( \vec{b} = (6, -8) \).

Step 2: Understand the Perpendicular Condition

For two vectors to be perpendicular, their dot product must equal zero. The dot product of two vectors \( \vec{a} = (a_1, a_2) \) and \( \vec{b} = (b_1, b_2) \) is given by \( a_1 \cdot b_1 + a_2 \cdot b_2 \).

Step 3: Calculate the Dot Product

Substitute the components of the vectors into the dot product formula: \((-4)(6) + (-3)(-8)\). This simplifies to \(-24 + 24\).

Step 4: Evaluate the Result

Calculate \(-24 + 24\), which equals 0. Since the result of the dot product is 0, the vectors are perpendicular.

Key Concepts

Dot Product

To determine if two vectors are perpendicular, we check their dot product. The dot product is a way to multiply two vectors to see how much they "align" with each other. If two vectors are perpendicular, they form a 90-degree angle, meaning their alignment is zero and thus their dot product also equals zero.
The dot product for two-dimensional vectors is calculated using the formula:

• For vectors \( \vec{a} = (a_1, a_2) \) and \( \vec{b} = (b_1, b_2) \), the dot product is \( a_1 \cdot b_1 + a_2 \cdot b_2 \).
• In the given problem, the calculation involves multiplying the corresponding components of the vectors \( (-4 \cdot 6) + (-3 \cdot -8) \).
• Simplifying this signifies \( -24 + 24 \) which gives \( 0 \), indicating that the vectors are perpendicular.

Understanding the calculation of the dot product is key in determining vector relationships like perpendicularity.

Vector Representation

Vectors are often represented in terms of their components along axes, typically as combinations of unit vectors like \( \mathbf{i} \) and \( \mathbf{j} \). This representation helps to visualize how vectors span directions in space. A vector \( \vec{v} \) expressed as \( a \mathbf{i} + b \mathbf{j} \) has components \( (a, b) \).
Considering the provided vectors:

• \(-4 \mathbf{i} - 3 \mathbf{j}\) becomes \((-4, -3)\).
• \(6 \mathbf{i} - 8 \mathbf{j}\) turns into \((6, -8)\).
• This notation helps us easily compute products, distances, and angles, crucial for geometric interpretations.

Seeing vectors as ordered pairs \( (a, b) \) allows us to leverage mathematical operations like addition, subtraction, and importantly, the dot product.

Vector Components

Understanding vector components is crucial for operations like the dot product. Each vector can be broken down into parts that specify its direction and magnitude along each axis. These parts are called vector components, and they indicate how much of each unity direction \( \mathbf{i} \) and \( \mathbf{j} \) is in the vector.
Key points to remember about vector components are:

• They help describe the vector's directional influence in a plane.
• By representing vectors such as \( (a, b) \), each number indicates the vector's impact along the horizontal (\( a \)) and vertical (\( b \)) axes.
• To calculate interactions, such as dot products between vectors, recognize that each component multiplies with its corresponding partner, summing the results.

Effective use of vector components allows us to methodically explore vector relationships, capabilities in transformations, and ultimately, geometric positioning in space.