Problem 31 - Trigonometry

Question

Show that each pair of vectors is perpendicular. \(\mathbf{i}+\mathbf{j}\) and \(\mathrm{i}-\mathrm{j}\) \(i\) and \(j\)

Step-by-Step Solution

Answer

Both pairs of vectors are perpendicular since their dot products are zero.

Step 1: Understand Perpendicular Vectors

Two vectors are perpendicular if their dot product is zero. We will calculate the dot product of the given vectors to determine if they are perpendicular.

Step 2: Define the Vectors

The vectors are given as \[ \mathbf{v}_1 = \mathbf{i} + \mathbf{j} \quad \text{and} \quad \mathbf{v}_2 = \mathbf{i} - \mathbf{j} \] Another pair is\[\mathbf{v}_3 = \mathbf{i} \quad \text{and} \quad \mathbf{v}_4 = \mathbf{j} \]

Step 3: Calculate the Dot Product for the First Pair

Compute the dot product of \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \):\[ (\mathbf{i} + \mathbf{j}) \cdot (\mathbf{i} - \mathbf{j}) = (\mathbf{i} \cdot \mathbf{i}) + (\mathbf{i} \cdot (-\mathbf{j})) + (\mathbf{j} \cdot \mathbf{i}) + (\mathbf{j} \cdot (-\mathbf{j})) \]Simplify using \( \mathbf{i} \cdot \mathbf{i} = 1 \), \( \mathbf{i} \cdot \mathbf{j} = 0 \), and \( \mathbf{j} \cdot \mathbf{j} = 1 \): \[ 1 + 0 + 0 - 1 = 0 \] The dot product is zero; thus, \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) are perpendicular.

Step 4: Calculate the Dot Product for the Second Pair

Compute the dot product of \( \mathbf{v}_3 \) and \( \mathbf{v}_4 \):\[ \mathbf{i} \cdot \mathbf{j} = 0 \]The dot product is zero, confirming that \( \mathbf{v}_3 \) and \( \mathbf{v}_4 \) are also perpendicular.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It helps determine the angle between two vectors and can tell us if they are perpendicular.
One of the key properties of the dot product is that if the result is zero, the two vectors are perpendicular to one another. To calculate the dot product, multiply the respective components of the two vectors and then sum them all up. For instance, if you have vectors \( \mathbf{a} = [a_1, a_2] \) and \( \mathbf{b} = [b_1, b_2] \), their dot product is \(a_1b_1 + a_2b_2\).
Here is something useful:

• Keep in mind: the effect of squaring the magnitudes of unit vectors.
• The resulting dot product highlights the geometric relationship between the vectors.

Using these calculations, small steps in algebra can reveal interesting conclusions about the nature of vectors.

Vector Operations

Vector operations like addition, subtraction, and the dot product are key components in understanding the behavior of vectors. These operations are not only tools for calculation but for understanding how vectors interact.
Addition and subtraction involve combining or removing vector components. For instance, if you have \( \mathbf{v}_1 = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v}_2 = \mathbf{i} - \mathbf{j} \), adding them gives you \( (\mathbf{i} + \mathbf{j}) + (\mathbf{i} - \mathbf{j}) = 2\mathbf{i} \). Here, the \( \mathbf{j} \) components cancel each other out.
Subtraction reverses this process and removes parts of the vector, aligning in an opposite manner. Practicing these operations, especially in dot product form, you can deduce spatial interactions between different vectors.
Overall:

• Vector operations bring clarity in physics and advanced mathematics.
• They are crucial for constructing calculations and for vector manipulation in real-world applications.

Unit Vectors

Unit vectors are the building blocks of vector spaces. They have a magnitude of 1 and are used to denote direction.
Common unit vectors in two-dimensional spaces include \( \mathbf{i} \) and \( \mathbf{j} \), where \( \mathbf{i} \) represents the x-direction and \( \mathbf{j} \) the y-direction.
The wonderful thing about unit vectors is their ability to simplify complex vector operations:

• They can be scaled by any magnitude without affecting direction.
• They serve as a reference for other vectors and easily reveal perpendicular relationships due to the zero dot product in perpendicular cases.

The essence of unit vectors lies in their simplicity, and they are fundamental in transforming an abstract concept like vectors into comprehensible math problems.