Problem 26 - Algebra and Trigonometry

Question

Use the dot product to determine whether v and w are orthogonal. $$ \mathbf{v}=8 \mathbf{i}-4 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}-12 \mathbf{j} $$

Step-by-Step Solution

Answer

Yes, the vectors \(v\) and \(w\) are orthogonal.

Step 1: Write down the vectors in component form

The vectors are written as \(v = (8, -4)\) and \(w = (-6, -12)\)

Step 2: Compute the dot product

The dot product of two vectors can be calculated as follows: \(v \cdot w = v_1w_1 + v_2w_2\). Substituting the components of v and w gives \( v \cdot w = 8*(-6) + (-4)*(-12)\)

Step 3: Check if the dot product equals zero

The dot product of v and w is calculated to be -48 + 48 = 0. Thus, the two vectors are orthogonal, because their dot product equals zero.

Key Concepts

Dot Product

The dot product, also referred to as the scalar product, is a fundamental operation in linear algebra. It is used to calculate a single number from two equal-length sequences of numbers, usually coordinate vectors. When dealing with vectors \( \mathbf{v} = (a, b) \) and \( \mathbf{w} = (c, d) \), the dot product is defined as: \[ \mathbf{v} \cdot \mathbf{w} = a \cdot c + b \cdot d \] This formula results in a scalar that gives an indication of the magnitude and direction similarity of the two vectors.

• If the dot product is zero, the vectors are orthogonal, meaning they are perpendicular to each other.
• This property is particularly useful in geometry and physics, where understanding the orientation of vectors can assist in solving many types of problems.

Understanding how to compute the dot product and interpret its result is key in many vector operations.

Vector Components

Vectors, which are mathematical entities with both magnitude and direction, can be broken down into components. These components reflect the vector's effect in multi-dimensional spaces and are usually represented in terms of unit vectors, like \( \mathbf{i} \) and \( \mathbf{j} \). For example, a vector \( \mathbf{v} = 8 \mathbf{i} - 4 \mathbf{j} \) can be expressed in component form as \( (8, -4) \).

• The first component \( 8 \) indicates how much the vector travels along the horizontal axis (\( \mathbf{i} \)).
• The second component \( -4 \) describes the movement along the vertical axis (\( \mathbf{j} \)).

Knowing vector components helps in performing operations like the dot product or assessing vector effects in different dimensions, aiding in the graphical interpretation of vectors.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. Its principles are extremely powerful and versatile, making them widely applicable in various fields such as engineering, physics, computer science, and economics. Some core concepts of linear algebra include:

• Vectors: Objects that contain both magnitude and direction.
• Matrix operations: Techniques to handle systems of equations or linear mappings efficiently.
• Determinants and eigenvalues: Used to solve linear equations, optimize solutions, or understand matrix properties.

Understanding how vectors interact through operations like the dot product is essential for grasping the broader concepts of linear algebra.

Vector Operations

Vector operations are operations that involve the manipulation of vectors to derive new vector expressions or quantities. Among the most common operations are addition, subtraction, scalar multiplication, and the dot product.

Common Vector Operations:

• Vector Addition: Combine components of two vectors to get a new vector.
• Vector Subtraction: Subtract components of one vector from another.
• Scalar Multiplication: Multiply each component of a vector by a scalar (number).
• Dot Product: Produce a scalar from two vectors, indicating their directional relationship.

These operations are a foundation of many scientific computations and are essential tools for anyone studying physics, engineering, computer graphics, or any field that involves dynamics and forces. Understanding these operations enables you to solve complex problems by breaking them down into more manageable parts.