Problem 16 - Trigonometry

Question

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=6 \mathbf{i}, \mathbf{V}=-8 \mathbf{j}\)

Step-by-Step Solution

Answer

The dot product \( \mathbf{U} \cdot \mathbf{V} \) is 0.

Step 1: Understand Dot Product Formula

The dot product between two vectors \( \mathbf{U} \) and \( \mathbf{V} \) is calculated using the formula \( \mathbf{U} \cdot \mathbf{V} = U_x V_x + U_y V_y + U_z V_z \) where \( U_x, U_y, U_z \) are the components of \( \mathbf{U} \) and \( V_x, V_y, V_z \) are those of \( \mathbf{V} \).

Step 2: Extract Vector Components

For \( \mathbf{U} = 6 \mathbf{i} \), the components are \( U_x = 6 \), \( U_y = 0 \), \( U_z = 0 \). For \( \mathbf{V} = -8 \mathbf{j} \), the components are \( V_x = 0 \), \( V_y = -8 \), \( V_z = 0 \).

Step 3: Apply Dot Product Formula

Substitute the extracted components into the dot product formula: \( \mathbf{U} \cdot \mathbf{V} = 6 \times 0 + 0 \times (-8) + 0 \times 0 \).

Step 4: Simplify the Expression

Calculate the result of the expression: \( \mathbf{U} \cdot \mathbf{V} = 0 + 0 + 0 = 0 \).

Key Concepts

Vectors

Vectors are fundamental objects in mathematics and physics, commonly used to represent quantities that have both magnitude and direction. Unlike scalars, which have only magnitude, vectors allow us to encapsulate directional information. You can visualize vectors as arrows having a specific length and pointing in a particular direction. This is useful for representing things like forces, velocities, or any directional quantity.

A vector is commonly denoted by a bold letter or an arrow above the letter, such as \( \mathbf{U} \) or \( \vec{U} \). Its components are typically written in relation to coordinate units like the \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) unit vectors, which represent the x, y, and z axes respectively. This allows for a straightforward representation of vectors in two or three-dimensional space.

• \( \mathbf{i} \) is the unit vector along the x-axis.
• \( \mathbf{j} \) is the unit vector along the y-axis.
• \( \mathbf{k} \) is the unit vector along the z-axis.

By breaking a vector into these components, we can perform calculations that involve direction and magnitude efficiently.

Vector Components

The components of a vector represent its projections along the coordinate axes. These components are essential when working with vectors, as they translate the abstract concept into tangible values that we can manipulate mathematically. For instance, if you have a vector \( \mathbf{U} \) with the equation \( \mathbf{U} = 6 \mathbf{i} \), this means that the vector has 6 units of magnitude in the positive x-direction and zero in the other directions.

Understanding vector components is crucial because they provide a structured way to perform vector addition, subtraction, and other vector operations. In a three-dimensional space, any vector can be broken down into three components corresponding to the \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) directions.

• If \( \mathbf{U} = U_x \mathbf{i} + U_y \mathbf{j} + U_z \mathbf{k} \), then the components are \( U_x \), \( U_y \), and \( U_z \).
• Each component represents how far the vector extends along each axis.
• Components are crucial for computing the dot product and cross product, which hinge on these individual directional measures.

Vector Operations

Vector operations include a variety of mathematical procedures that can be performed on vectors, such as addition, subtraction, scalar multiplication, and the dot and cross products. These operations allow for complex vector manipulations necessary in physics, engineering, computer graphics, and more.

The dot product, in particular, is a scalar representation of the vector multiplication, providing insight into the relationship between two vectors in terms of direction. When calculating the dot product, you use the vector components to derive a single scalar value. The dot product formula for vectors \( \mathbf{U} \) and \( \mathbf{V} \) is:
\[ \mathbf{U} \cdot \mathbf{V} = U_x V_x + U_y V_y + U_z V_z \]

From the example of \( \mathbf{U}=6 \mathbf{i} \) and \( \mathbf{V}=-8 \mathbf{j} \), the dot product calculation would yield zero, highlighting one of its significant properties: when two vectors are perpendicular, their dot product is zero.

• Dot product simplifications depend heavily on the components being zero or opposite in direction.
• It provides a way to determine angles between vectors, with zero indicating perpendicularity.
• Vector operations form the backbone of great many applications in scientific computations and real-world problem-solving.