Problem 30 - Algebra and Trigonometry

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: Identify the Vectors

Firstly, we need to identify the components of each vector. The first vector is given as \( 4 \mathbf{j} - \mathbf{k} \), which can be represented as \( (0, 4, -1) \). The second vector is given as \( \mathbf{i} + 2 \mathbf{j} + 9 \mathbf{k} \), which can be written as \( (1, 2, 9) \).

Step 2: Calculate the Dot Product

To determine if the vectors are perpendicular, calculate their dot product. The formula for the dot product in three dimensions is: \[ a_1 \times b_1 + a_2 \times b_2 + a_3 \times b_3 \].

Step 3: Substitute and Calculate Dot Product

Substitute the values of the components into the dot product formula: \[ 0 \times 1 + 4 \times 2 + (-1) \times 9 = 0 + 8 - 9 = -1 \].

Step 4: Determine Perpendicularity

A dot product of zero indicates that two vectors are perpendicular. Since the calculated dot product is \(-1\), the vectors are not perpendicular.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics that helps determine the relationship between two vectors, especially in terms of their direction. The dot product, also known as the scalar product, results in a single number obtained from the sum of the products of the corresponding components of two vectors.
This can be expressed using the formula:

• If vector \(\text{**a**} = (a_1, a_2, a_3)\) and vector \(\text{**b**} = (b_1, b_2, b_3)\), then their dot product is \(\text{**a**} \cdot \text{**b**} = a_1 \times b_1 + a_2 \times b_2 + a_3 \times b_3\).

A key property of the dot product is that it indicates whether two vectors are perpendicular (orthogonal) when the result is zero. This is because a zero dot product implies there is no projection of one vector onto another. If the vectors were given as in the exercise and the dot product resulted in zero, these vectors would have been perpendicular. In the given exercise, however, the dot product was \(-1\), confirming the vectors are not perpendicular.

Vector Components

Vector components give a clear picture of the direction and magnitude of vectors in different dimensions. Any vector in three-dimensional space can be broken down into components that correspond to the axis directions, typically labeled as \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\).
These components represent:

• The \(\mathbf{i}\) component (usually the x-axis).
• The \(\mathbf{j}\) component (usually the y-axis).
• The \(\mathbf{k}\) component (usually the z-axis).

In the exercise example, the first vector \((0, 4, -1)\) has components indicating it moves along the y and z axes while the second vector \(1, 2, 9\) spans all three axes. Representing vectors in terms of their components simplifies operations like the dot product and helps visualize their orientation and effect in space.

Three-Dimensional Vectors

Three-dimensional vectors offer a versatile way of representing quantities with both direction and magnitude in a three-dimensional space. They are used in a variety of contexts, from physics to computer graphics, to simulate real-world phenomena.
Vectors are usually denoted as \(\mathbf{v} = (v_1, v_2, v_3)\), where each entry corresponds to a magnitude in the direction of any of the space's three dimensions.
This enables the representation of physical quantities like velocity, force, and displacement.

• These vectors obey specific algebraic rules and can be added, subtracted, or scaled, making them integral to geometric interpretations.
• Operations like the dot product in three dimensions help to find angles between vectors and assess perpendicularity, as used in the given exercise problem.

Three-dimensional vectors allow models and calculations that involve immersive, dynamic environments to be constructed and analyzed seamlessly.