Problem 36 - Trigonometry

Question

Show that each pair of vectors is perpendicular. Find the value of \(a\) so that vectors \(\mathbf{U}=a \mathbf{i}+6 \mathbf{j}\) and \(\mathbf{V}=9 \mathbf{i}+12 \mathbf{j}\) are perpendicular

Step-by-Step Solution

Answer

The vectors are perpendicular when \(a = -8\).

Step 1: Understand Perpendicular Vectors

Two vectors are perpendicular if their dot product is equal to zero. For vectors \(\mathbf{U}=a \mathbf{i}+6 \mathbf{j}\) and \(\mathbf{V}=9 \mathbf{i}+12 \mathbf{j}\), we need to set up the dot product formula and set it equal to zero.

Step 2: Write the Dot Product Expression

The dot product of vectors \(\mathbf{U}\) and \(\mathbf{V}\) is calculated by multiplying their corresponding components and summing the results. Therefore, the dot product is: \(a \times 9 + 6 \times 12\).

Step 3: Solve the Dot Product Equation

Set the dot product expression equal to zero: \(9a + 72 = 0\). Solve for \(a\) by first subtracting 72 from both sides: \(9a = -72\).

Step 4: Find the Value of \(a\)

Divide both sides by 9 to isolate \(a\): \(a = -8\). This is the value of \(a\) that makes the vectors perpendicular.

Key Concepts

Dot Product

The dot product is a crucial concept in vector mathematics, especially when analyzing perpendicular vectors. It is a way of multiplying two vectors to obtain a scalar. To compute the dot product of two vectors, you multiply their corresponding components and sum the results.
For example, given two vectors \( \mathbf{U} = a \mathbf{i} + 6 \mathbf{j} \) and \( \mathbf{V} = 9 \mathbf{i} + 12 \mathbf{j} \), their dot product is calculated as follows:

• Multiply the \( \mathbf{i} \)-components: \( a \times 9 \)
• Multiply the \( \mathbf{j} \)-components: \( 6 \times 12 \)
• Sum the results: \( 9a + 72 \)

This expression represents the dot product of vectors \( \mathbf{U} \) and \( \mathbf{V} \). For two vectors to be perpendicular, their dot product must be zero.
This is because zero indicates no component of one vector in the direction of the other, meaning they are orthogonal or perpendicular.

Vector Components

Vector components are the building blocks of vectors in a coordinate system. They represent how much of each unit vector (\( \mathbf{i} \) and \( \mathbf{j} \) in two-dimensional space) makes up the vector. Consider a vector \( \mathbf{U} = a \mathbf{i} + 6 \mathbf{j} \):

• The \( \mathbf{i} \)-component is \( a \), indicating the horizontal part of the vector.
• The \( \mathbf{j} \)-component is \( 6 \), indicating the vertical part of the vector.

Vector \( \mathbf{V} = 9 \mathbf{i} + 12 \mathbf{j} \) is similar, with components:

• \( 9 \mathbf{i} \) as the horizontal or \( \mathbf{i} \)-component.
• \( 12 \mathbf{j} \) as the vertical or \( \mathbf{j} \)-component.

Understanding vector components is essential for performing vector operations like the dot product. The components also help identify directions and magnitudes within a coordinate plane.
Recognizing how vectors are broken down into \( \mathbf{i} \) and \( \mathbf{j} \) components is vital for solving equations involving vectors.

Solving Equations

Solving equations is an integral mathematical skill, especially when dealing with vectors and their properties. In the context of finding when two vectors are perpendicular, we use the equation derived from their dot product and solve it.
In our example, we set the dot product equal to zero to reflect the condition for perpendicularity:

• Let the dot product equation be \( 9a + 72 = 0 \).
• Subtract 72 from both sides to simplify: \( 9a = -72 \).
• Divide both sides by 9 to solve for \( a \): \( a = -8 \).

This process of manipulating equations involves basic algebraic steps such as moving terms and dividing by coefficients. These operations make it possible to find the unknown variable that fulfills the perpendicularity condition.
Practicing these steps builds the foundational skills necessary for more advanced algebra and calculus.