https://file.aiquickdraw.com/mathbot/file/1726196392368jrkzaj7e.jpg

The problem asks to compute two things based on two vectors:

Given:
\[
\vec{A} = 4.00\hat{i} + 7.00\hat{j}
\]
and
\[
\vec{B} = 5.00\hat{i} - 2.00\hat{j}
\]

### Part A: Finding the cross product \(\vec{A} \times \vec{B}\)

The cross product formula for two vectors \(\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) is:

\[
\vec{A} \times \vec{B} = 
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3
\end{vmatrix}
\]

Since \(\vec{A}\) and \(\vec{B}\) do not have a \(\hat{k}\) component (i.e., both are 2D vectors in the xy-plane), we can simplify by assuming \(a_3 = b_3 = 0\):

\[
\vec{A} \times \vec{B} = \hat{i}(0) - \hat{j}(0) + \hat{k}((4)(-2) - (7)(5))
\]

\[
\vec{A} \times \vec{B} = \hat{k}((-8) - 35) = \hat{k}(-43)
\]

Thus, the vector cross product is:
\[
\vec{A} \times \vec{B} = -43\hat{k}
\]

### Part B: Magnitude of the vector product

The magnitude of the cross product is given by:

\[
|\vec{A} \times \vec{B}| = \sqrt{(-43)^2} = 43
\]

### Final answers:

- Part A: \(\vec{A} \times \vec{B} = -43\hat{k}\)
- Part B: \(|\vec{A} \times \vec{B}| = 43\)

Would you like more details or have any questions?

Here are some related questions:

1. What is the geometric interpretation of a cross product?
2. How would the result change if one of the vectors had a \(\hat{k}\) component?
3. How do you compute the cross product for 3D vectors?
4. Can you explain the right-hand rule for determining the direction of the cross product?
5. How would you calculate the dot product of the same vectors?

**Tip:** The cross product is perpendicular to the plane formed by the two vectors, and its magnitude represents the area of the parallelogram they form.

Given two vectors A = 4.00i + 7.00j and B = 5.00i - 2.00j, find the vector product A × B (expressed in unit vectors) and calculate its magnitude.

Learn how to find the cross product of two vectors A = 4.00i + 7.00j and B = 5.00i - 2.00j. This example walks through the calculation of the vector product and its magnitude using vector calculus concepts. Suitable for advanced high school or college 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