Problem 37 - Trigonometry

Question

Show that each pair of vectors is perpendicular. In general, show that the vectors \(\mathbf{V}=a \mathbf{i}+b \mathbf{j}\) and \(\mathbf{W}=-b \mathbf{i}+a \mathbf{j}\) are always perpendicular. Assume \(a\) and \(b\) are not both equal to zero.

Step-by-Step Solution

Answer

Vectors \( \mathbf{V} = a \mathbf{i} + b \mathbf{j} \) and \( \mathbf{W} = -b \mathbf{i} + a \mathbf{j} \) are perpendicular. Their dot product is zero.

Step 1: Understanding Vector Perpendicularity

Two vectors are perpendicular if their dot product is zero. The dot product formula for vectors \( \mathbf{V} = a \mathbf{i} + b \mathbf{j} \) and \( \mathbf{W} = -b \mathbf{i} + a \mathbf{j} \) is \( \mathbf{V} \cdot \mathbf{W} = 0 \).

Step 2: Write the Dot Product Expression

The dot product of vectors \( \mathbf{V} = a \mathbf{i} + b \mathbf{j} \) and \( \mathbf{W} = -b \mathbf{i} + a \mathbf{j} \) is calculated as: \[ \mathbf{V} \cdot \mathbf{W} = (a \mathbf{i} + b \mathbf{j}) \cdot (-b \mathbf{i} + a \mathbf{j}) \].

Step 3: Compute Each Component of the Dot Product

Compute the dot product step by step: * The \( \mathbf{i} \) components: \( a \cdot (-b) = -ab \). * The \( \mathbf{j} \) components: \( b \cdot a = ab \).

Step 4: Combine the Components

Combine the results of components from Step 3: \( \mathbf{V} \cdot \mathbf{W} = -ab + ab \).

Step 5: Simplify the Expression

Simplify the combined dot product to see if it equals zero: \( \mathbf{V} \cdot \mathbf{W} = -ab + ab = 0 \).

Step 6: Verify the Result

Since \( \mathbf{V} \cdot \mathbf{W} = 0 \), this shows that vectors \( \mathbf{V} \) and \( \mathbf{W} \) are indeed perpendicular.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics, used to find the relationship between two vectors. For two-dimensional vectors, the dot product is computed by multiplying corresponding components and adding the results. For example, if we have two vectors \(\mathbf{V} = a \mathbf{i} + b \mathbf{j}\) and \(\mathbf{W} = c \mathbf{i} + d \mathbf{j}\), their dot product is given as:\[ \mathbf{V} \cdot \mathbf{W} = ac + bd\]This operation results in a scalar quantity rather than a vector. It provides crucial information about the angle between the vectors. In particular, if the dot product equals zero, the vectors are perpendicular (orthogonal). This is because the cosine of the angle between them becomes zero, which occurs at 90 degrees or \(\frac{\pi}{2}\) radians. Thus, when solving problems involving vector perpendicularity, the core idea is to find the dot product and check if it is zero to confirm that the vectors are perpendicular.

Vector Components

Vector components are the building blocks of vectors. They allow us to separate a vector into parts that align with the coordinate axes we are using, typically the \(\mathbf{i}\) and \(\mathbf{j}\) units in a two-dimensional space. This separation is essential for calculations like the dot product and assessing vector relationships.A vector such as \(\mathbf{V} = a \mathbf{i} + b \mathbf{j}\) consists of two components:

• \(a\) along the x-axis direction (represented by the unit vector \(\mathbf{i}\)),
• \(b\) along the y-axis direction (represented by the unit vector \(\mathbf{j}\)).

Each component describes how much of the vector extends in a particular direction. To understand a vector geometrically, considering its components allows us to visualize it as a combination of its parts along the coordinate axes. This understanding simplifies many problems in vector calculus, as it breaks down complex relationships into manageable pieces. It is also crucial for forming the dot product, as each component multiplication contributes to the final scalar result.

Vector Perpendicularity

Perpendicular vectors are vectors that intersect at a right angle. In a two-dimensional plane, this right angle is 90 degrees. One of the most straightforward methods to determine if two vectors \(\mathbf{V}\) and \(\mathbf{W}\) are perpendicular is to use the dot product. The perpendicularity condition is given by:\[ \mathbf{V} \cdot \mathbf{W} = 0 \]When this condition holds, it indicates that the vectors do not share any components in the same direction. For example, for vectors \(\mathbf{V} = a \mathbf{i} + b \mathbf{j}\) and \(\mathbf{W} = -b \mathbf{i} + a \mathbf{j}\), calculating the dot product gives zero as we showed in the step-by-step solution:\[ \mathbf{V} \cdot \mathbf{W} = -ab + ab = 0 \]This calculation proves the vectors are perpendicular, demonstrating that neither vector has a component influencing the other in the shared dimensional space. Understanding this concept is essential as it forms the basis for many other vector operations and applications, such as projection and vector decomposition.