Problem 30 - Precalculus Mathematics for Calculus

Question

Determine whether or not the given vectors are perpendicular. $$4 \mathbf{j}-\mathbf{k}, \quad \mathbf{i}+2 \mathbf{j}+9 \mathbf{k}$$

Step-by-Step Solution

Answer

The vectors are not perpendicular.

Step 1: Understand the Condition for Perpendicular Vectors

Two vectors are perpendicular if their dot product is zero. Given the vectors \( \mathbf{A} = 4\mathbf{j}-\mathbf{k} \) and \( \mathbf{B} = \mathbf{i}+2 \mathbf{j}+9 \mathbf{k} \), we will calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \).

Step 2: Express Vectors in Component Form

Write the vectors in component form: \( \mathbf{A} = [0, 4, -1] \) and \( \mathbf{B} = [1, 2, 9] \).

Step 3: Calculate the Dot Product

Compute the dot product \( \mathbf{A} \cdot \mathbf{B} = 0 \cdot 1 + 4 \cdot 2 + (-1) \cdot 9 \).

Step 4: Simplify the Dot Product

Calculate the multiplication and summation: \( 0 \cdot 1 = 0 \), \( 4 \cdot 2 = 8 \), and \( -1 \cdot 9 = -9 \). Therefore, \( \mathbf{A} \cdot \mathbf{B} = 0 + 8 - 9 \).

Step 5: Conclude Dot Product Evaluation

Evaluate the sum: \( \mathbf{A} \cdot \mathbf{B} = 0 + 8 - 9 = -1 \).

Step 6: Determine Perpendicularity

Since the dot product \( \mathbf{A} \cdot \mathbf{B} = -1 eq 0 \), the vectors are not perpendicular.

Key Concepts

Dot Product

In vector mathematics, the dot product is a crucial operation, especially when exploring the idea of perpendicularity between two vectors. The dot product of two vectors is the sum of the products of their corresponding components. Mathematically, for two vectors \( \mathbf{A} = [a_1, a_2, a_3] \) and \( \mathbf{B} = [b_1, b_2, b_3] \), the dot product is calculated as:

• \( \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \)

This operation is central to determining whether vectors are perpendicular, as perpendicular vectors satisfy the condition where their dot product equals zero.

Vector Components

Vectors in the Cartesian coordinate system have components, which are akin to coordinates. These components provide the vector's magnitude and direction. For example, the vector \( 4\mathbf{j}-\mathbf{k} \) can be expressed in component form as \( [0, 4, -1] \).
The components directly reflect the vector's influence along the respective axis:

• The \( i \)-component indicates horizontal movement along the x-axis.
• The \( j \)-component signifies vertical movement along the y-axis.
• The \( k \)-component points to movement along the z-axis.

Thus, writing vectors in component form is a fundamental step in operations like calculating the dot product.

Calculating Dot Product

The calculation of the dot product involves a straightforward application of multiplication and addition of vector components. Given vectors \( \mathbf{A} = [0, 4, -1] \) and \( \mathbf{B} = [1, 2, 9] \), the dot product is computed sequentially:

• Multiply corresponding components: \( 0 \times 1, 4 \times 2, \text{and} (-1) \times 9 \).
• Results in: \( 0, 8, \text{and} -9 \).
• Sum the results: \( 0 + 8 - 9 \).

This leads to a final dot product of \(-1\). Calculating the dot product accurately is key to determining perpendicularity.

Vector Perpendicularity

Perpendicularity of vectors is a geometric property denoting two vectors that intersect at a right angle (90 degrees). A unique attribute of perpendicular vectors is that their dot product is always zero. However, in the exercise, we found \( \mathbf{A} \cdot \mathbf{B} = -1 \), indicating that the vectors are not perpendicular.
Understanding this relationship and the importance of a zero dot product can be pivotal when analyzing vector alignments in fields such as physics and engineering.

Mathematical Vectors

Mathematical vectors are entities possessing both magnitude and direction, and are represented graphically by arrows. Their components correspond to each dimension in space where the vector is defined.
Vectors are represented by their components in mathematical problems:

• Position vectors describe location relative to an origin.
• Displacement vectors describe movement from one point to another.

They are fundamental in various mathematical and practical applications, from computer graphics to physical sciences, providing a powerful tool to describe and manipulate spatial relationships.