https://file.aiquickdraw.com/mathbot/file/17262414434666cnhxw21.jpg

In the image, we can see a two-part problem related to vectors.

### Part A:
You're asked to find the scalar (dot) product of two vectors \(\vec{A}\) and \(\vec{B}\), where:

\[
\vec{A} = 3.90\hat{i} + 6.90\hat{j}
\]
\[
\vec{B} = 5.30\hat{i} - 2.30\hat{j}
\]

The dot product of two vectors \(\vec{A} \cdot \vec{B}\) is calculated as:

\[
\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y
\]

Where:
- \( A_x = 3.90 \), \( A_y = 6.90 \)
- \( B_x = 5.30 \), \( B_y = -2.30 \)

Substituting these values:

\[
\vec{A} \cdot \vec{B} = (3.90)(5.30) + (6.90)(-2.30)
\]

\[
= 20.67 + (-15.87) = 4.80
\]

Which matches the correct answer shown: **4.80**.

---

### Part B:
You're asked to find the angle \(\theta\) between the two vectors. The formula to find the angle between two vectors is:

\[
\cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}
\]

Where:
- \(\vec{A} \cdot \vec{B} = 4.80\)
- \(|\vec{A}|\) is the magnitude of vector \(\vec{A}\), and \(|\vec{B}|\) is the magnitude of vector \(\vec{B}\).

The magnitude of a vector \(\vec{A} = A_x\hat{i} + A_y\hat{j}\) is:

\[
|\vec{A}| = \sqrt{A_x^2 + A_y^2} = \sqrt{(3.90)^2 + (6.90)^2}
\]

\[
|\vec{A}| = \sqrt{15.21 + 47.61} = \sqrt{62.82} \approx 7.93
\]

Similarly, for \(\vec{B} = B_x\hat{i} + B_y\hat{j}\):

\[
|\vec{B}| = \sqrt{(5.30)^2 + (-2.30)^2}
\]

\[
|\vec{B}| = \sqrt{28.09 + 5.29} = \sqrt{33.38} \approx 5.78
\]

Now using the dot product formula to find \(\theta\):

\[
\cos \theta = \frac{4.80}{(7.93)(5.78)} = \frac{4.80}{45.84} \approx 0.1047
\]

Thus:

\[
\theta = \cos^{-1}(0.1047) \approx 84.0^\circ
\]

This matches the given answer of **84.0°**.

---

Would you like more detailed explanations or have any other questions?

### Related Questions:
1. How do you calculate the cross product of two vectors in 3D space?
2. What are the properties of dot products and cross products?
3. How does the dot product differ from the cross product in terms of physical interpretation?
4. How do you find the projection of one vector onto another?
5. What does it mean geometrically when the dot product of two vectors is zero?

### Tip:
Always check the magnitudes of vectors when calculating angles to avoid errors, especially in cases involving unit vectors.

Given two vectors A = 3.90i + 6.90j and B = 5.30i - 2.30j, find the scalar product of the two vectors A and B. Also, find the angle between these two vectors.

Solve the scalar product of vectors A = 3.90i + 6.90j and B = 5.30i - 2.30j, and find the angle between them using dot product and magnitude formulas. Detailed step-by-step explanation with vector algebra and trigonometry concepts. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation