Problem 32 - Trigonometry

Question

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

Step-by-Step Solution

Answer

Yes, the vectors \(-\mathbf{i}\) and \(\mathbf{j}\) are perpendicular because their dot product is zero.

Step 1: Understand Perpendicular Vectors

Two vectors are perpendicular if their dot product equals zero. We will verify if the vectors given, \(-\mathbf{i}\) and \(\mathbf{j}\), are perpendicular by calculating the dot product and checking if it results in zero.

Step 2: Define the Vectors

The vector \(-\mathbf{i}\) corresponds to \((-1, 0)\) and the vector \(\mathbf{j}\) corresponds to \((0, 1)\). These are the standard unit vectors along the x-axis and y-axis, respectively, with \(-\mathbf{i}\) being the negative unit vector along the x-axis.

Step 3: Calculate the Dot Product

The dot product of two vectors \(\mathbf{a} = (a_1, a_2)\) and \(\mathbf{b} = (b_1, b_2)\) is calculated as follows: \(\mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2\). Apply this to the vectors \(-\mathbf{i}\) and \(\mathbf{j}\): \[(-1) \cdot 0 + 0 \cdot 1 = 0 + 0 = 0.\]

Step 4: Conclusion

Since the dot product of \(-\mathbf{i}\) and \(\mathbf{j}\) is zero, the two vectors are indeed perpendicular. This conclusion follows from the definition provided in Step 1.

Key Concepts

Dot Product

The dot product is an essential concept when analyzing vectors, especially when determining if they are perpendicular. In mathematics, two vectors are considered perpendicular if their dot product equals zero.
This means they form a right angle with each other in space. For any two vectors, \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), the dot product is calculated using:

• \( \mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 \)

The dot product tells us the degree to which two vectors "push" in the same direction. A zero result means there is no such push, indicating perpendicularly.
In our exercise, we've shown that the vectors \(-\mathbf{i}\) and \(\mathbf{j}\) have a dot product of zero. Thus, they are perpendicular as they form a right angle.

Unit Vectors

Unit vectors are vectors with a length of one. They are fundamental in vector mathematics because they represent directional information without concerning magnitude.
Typical unit vectors in a 2D Cartesian coordinate system are \( \mathbf{i} \) and \( \mathbf{j} \). Here:

• \( \mathbf{i} \) is the unit vector along the x-axis, represented as \((1, 0)\).
• \( \mathbf{j} \) is the unit vector along the y-axis, represented as \((0, 1)\).

When combined and scaled, unit vectors can create any other vector by determining its direction. The negativity in a unit vector, such as \(-\mathbf{i}\), simply means it points in the opposite direction along the x-axis.
In our problem, \(-\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors, with lengths of one and pointing in precisely perpendicular directions.

Vector Components

Vector components break down a vector into its constituent parts, typically in the x, y, and sometimes z directions in three-dimensional space.
This breakdown allows us to perform operations like addition, subtraction, and analyzing vector relationships. Each component tells us how much influence the vector has in a particular direction.
The basic notation for a 2D vector \( \mathbf{v} \) is:

• \( \mathbf{v} = (x, y) \)

Here, \( x \) is the vector's x-component, and \( y \) is the y-component. For instance, the vector \(-\mathbf{i}\) is \((-1, 0)\), indicating full movement in the negative x-direction and none in the y-direction. Similarly, \(\mathbf{j}\) is \((0, 1)\), moving completely in the positive y-direction.
Understanding these components makes it clear how operations like the dot product are performed, as each component individually contributes to the final result, such as determining the perpendicularity described in the exercise.