Problem 31 - Precalculus Mathematics for Calculus

Question

Determine whether or not the given vectors are perpendicular. $$(0.3,1.2,-0.9),(10,-5,10)$$

Step-by-Step Solution

Answer

The vectors are not perpendicular.

Step 1: Understand Perpendicular Vectors

Two vectors are perpendicular if their dot product equals zero. The dot product of two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \) is given by \( a_1b_1 + a_2b_2 + a_3b_3 \).

Step 2: Identify Components of Vectors

Identify the components of the vectors. For vector \( \mathbf{a} = (0.3, 1.2, -0.9) \), the components are \( a_1 = 0.3 \), \( a_2 = 1.2 \), \( a_3 = -0.9 \). For vector \( \mathbf{b} = (10, -5, 10) \), the components are \( b_1 = 10 \), \( b_2 = -5 \), \( b_3 = 10 \).

Step 3: Calculate Dot Product

Substitute the components into the dot product formula: \[ 0.3 \times 10 + 1.2 \times (-5) + (-0.9) \times 10 \] Calculate each term: \( 0.3 \times 10 = 3 \), \( 1.2 \times (-5) = -6 \), \( -0.9 \times 10 = -9 \).

Step 4: Sum the Results

Compute the sum of the results from Step 3: \[ 3 + (-6) + (-9) = 3 - 6 - 9 = -12 \] Check if the sum is zero to determine perpendicularity.

Step 5: Determine Perpendicularity

Since the dot product \(-12\) is not equal to zero, the vectors are not perpendicular.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics. It's a way to multiply two vectors, and the result is a scalar (a single number). This operation is denoted by a dot between the two vectors, like this:

• \( \mathbf{a} \cdot \mathbf{b} \).

For vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the dot product is computed by:

• Multiplying their corresponding components.
• Then summing these products: \( a_1b_1 + a_2b_2 + a_3b_3 \).

This is useful for determining angles between vectors. If the dot product is zero, it indicates that the vectors are perpendicular. This makes the dot product an essential tool for calculating vector relationships.

Perpendicular Vectors

Perpendicular vectors are two vectors that meet at a 90-degree angle. Determining if vectors are perpendicular is crucial in physics and engineering.This condition is identifiable through their dot product.

• If the dot product is zero, the vectors are perpendicular.

This happens because a zero dot product implies the cosine of the angle between them is zero, which corresponds to an angle of 90 degrees.In the example with vectors \( (0.3, 1.2, -0.9) \) and \( (10, -5, 10) \), the dot product was calculated as:

• \( 0.3 \times 10 + 1.2 \times (-5) + (-0.9) \times 10 = -12 \).

This shows that they are not perpendicular as the product is not zero, hence, they do not form a right angle.

Vector Components

Vectors have components that define their direction and magnitude in each dimension. Consider a vector in three-dimensional space \( (a_1, a_2, a_3) \), where each number is a component.

• The first component \( a_1 \) indicates direction along the x-axis.
• The second component \( a_2 \) along the y-axis.
• The third component \( a_3 \) along the z-axis.

Knowing these components helps in various calculations like the dot product. To examine vectors \( (0.3, 1.2, -0.9) \) and \( (10, -5, 10) \), each component's multiplication is part of computing the dot product to check perpendicularity:

• Components of vector \( \mathbf{a} \): \( 0.3, 1.2, -0.9 \)
• Components of vector \( \mathbf{b} \): \( 10, -5, 10 \)

These elements make the fundamentals in analyzing the vectors' relationships and performing computations like dot products.