Problem 67 - Fundamentals of Physics

Question

Let \(\hat{i}\) be directed to the east, \(\hat{\mathrm{j}}\) be directed to the north, and \(\hat{\mathrm{k}}\) be directed upward. What are the values of products (a) \(\hat{i} \cdot \hat{k},(b)\) \((-\hat{\mathrm{k}}) \cdot(-\hat{\mathrm{j}})\), and \((\mathrm{c}) \hat{\mathrm{j}} \cdot(-\hat{\mathrm{j}}) ?\) What are the directions (such as east or down of products \((\mathrm{d}) \hat{\mathrm{k}} \times \hat{\mathrm{j}},(\mathrm{e})(-\hat{\mathrm{i}}) \times(-\hat{\mathrm{j}})\), and \((\mathrm{f})(-\hat{\mathrm{k}}) \times(-\hat{\mathrm{j}}) ?\)

Step-by-Step Solution

Answer

(a) 0, (b) 0, (c) -1; (d) west, (e) up, (f) west.

Step 1: Dot Product of Perpendicular Unit Vectors

The dot product of two perpendicular unit vectors is zero. Since \( \hat{i} \) (east) and \( \hat{k} \) (up) are perpendicular, we compute \( \hat{i} \cdot \hat{k} = 0 \).

Step 2: Dot Product of Negative Unit Vectors

When computing the dot product of negative unit vectors, the negative signs cancel out. Thus, \((-\hat{k}) \cdot (-\hat{j}) = \hat{k} \cdot \hat{j}\). Since \( \hat{k} \) and \( \hat{j} \) (north and up) are perpendicular, the result is \(0\).

Step 3: Dot Product of Opposite Directions

The dot product of a unit vector and its negative is \(-1\) because they are in opposite directions. Therefore, \( \hat{j} \cdot (-\hat{j}) = -1\).

Step 4: Cross Product of \( \hat{k} \) and \( \hat{j} \)

The cross product results in a vector perpendicular to both, following the right-hand rule. \( \hat{k} \times \hat{j} = -\hat{i} \), pointing west.

Step 5: Cross Product of Negative Unit Vectors in New Plane

When calculating \((-\hat{i}) \times (-\hat{j})\), the negatives cancel, giving \(\hat{i} \times \hat{j} = \hat{k}\). Thus, it points up.

Step 6: Cross Product of \(-\hat{k}\) and \(-\hat{j}\)

Using the right-hand rule for \(-\hat{k} \times -\hat{j}\), since the negatives cancel, we find \(\hat{k} \times \hat{j}\). As in Step 4, this points west, giving \(-\hat{i}\).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation for vectors. It involves two vectors and results in a scalar value. Understanding the dot product can help in various vector calculations. Here are some important characteristics:

• The dot product is computed by multiplying corresponding components of two vectors and summing the results. For example, given vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\), their dot product is \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\).
• For unit vectors, the dot product is simply the cosine of the angle between them multiplied by the magnitudes of both vectors. Since unit vectors have a magnitude of 1, the result is the cosine of the angle between them.
• When vectors are perpendicular, their dot product is zero because the cosine of 90 degrees is zero. This explains why \(\hat{i} \cdot \hat{k} = 0\) and \(\hat{k} \cdot \hat{j} = 0\) in the exercise.
• The dot product of a vector with itself or its negative results in 1 or -1, respectively. This is due to the angle being 0 or 180 degrees. Therefore, \(\hat{j} \cdot (-\hat{j}) = -1\).

Cross Product

The cross product, unlike the dot product, results in a vector rather than a scalar. The cross product of two vectors is a vector that is perpendicular to the plane formed by the original vectors. Here are key aspects to consider for the cross product:

• The magnitude of the cross product of vectors \(\mathbf{a}\) and \(\mathbf{b}\) is given by \(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}| \sin \theta\), where \(\theta\) is the angle between them.
• The direction of the result follows the right-hand rule. Place your right hand so that your thumb points in the direction of the first vector \(\mathbf{a}\), and then curl your fingers toward the second vector \(\mathbf{b}\). Your thumb then points in the direction of the cross product.
• For unit vectors like \(\hat{k} \times \hat{j}\), it results in another unit vector \(-\hat{i}\), indicating a direction to the west, following the right-hand rule.
• Cross products involving negative unit vectors can simplify calculations, as the negative signs typically cancel. For example, \((-\hat{i}) \times (-\hat{j})\) simplifies to \(\hat{i} \times \hat{j}\), which points up or in the direction of \(\hat{k}\).

Unit Vectors

Unit vectors are vectors with a magnitude of 1. They serve as the building blocks for defining vector directions in space. Understanding unit vectors is crucial for both the dot and cross products. Let's delve into what makes unit vectors special:

• A unit vector indicates direction without regard to magnitude. For example, \(\hat{i}\) points east, \(\hat{j}\) points north, and \(\hat{k}\) points upward.
• Unit vectors simplify vector calculations, acting as the foundation for expressing any vector in terms of its components. If a vector \(\mathbf{v}\) is expressed as \(v_1\hat{i} + v_2\hat{j} + v_3\hat{k}\), then it is a combination of unit vectors in specific proportions.
• In the context of vector operations, unit vectors are convenient for performing dot and cross products, as their magnitude is 1. This greatly simplifies many calculations, as seen in this exercise.
• Utilizing unit vectors clarifies the application of both dot and cross products by providing a clear reference for direction and position in three-dimensional space.