Join the IDNLearn.com community and get your questions answered by knowledgeable individuals. Our experts are ready to provide prompt and detailed answers to any questions you may have.
Sagot :
Answer:To determine the scalar product (dot product) of the vectors \(\mathbf{A} = 6\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}\) and \(\mathbf{B} = 5\mathbf{i} - 6\mathbf{j} - 3\mathbf{k}\), we use the formula for the dot product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\):
\[
\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z
\]
Where \(\mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k}\) and \(\mathbf{B} = B_x \mathbf{i} + B_y \mathbf{j} + B_z \mathbf{k}\).
Given:
\[
\mathbf{A} = 6\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}
\]
\[
\mathbf{B} = 5\mathbf{i} - 6\mathbf{j} - 3\mathbf{k}
\]
The components are:
\[
A_x = 6, \quad A_y = 4, \quad A_z = -2
\]
\[
B_x = 5, \quad B_y = -6, \quad B_z = -3
\]
Now we calculate the dot product:
\[
\mathbf{A} \cdot \mathbf{B} = (6 \times 5) + (4 \times -6) + (-2 \times -3)
\]
Performing the multiplications:
\[
6 \times 5 = 30
\]
\[
4 \times -6 = -24
\]
\[
-2 \times -3 = 6
\]
Adding these results together:
\[
30 + (-24) + 6 = 30 - 24 + 6 = 12
\]
Thus, the scalar product of \(\mathbf{A}\) and \(\mathbf{B}\) is 12.
### Answer:
C. 12
Step-by-step explanation:Certainly! Let's go through the steps to find the scalar product (dot product) of vectors \( A = 6i + 4j - 2k \) and \( B = 5i - 6j - 3k \).
The formula for the scalar product (dot product) of two vectors \( A = (a_1, a_2, a_3) \) and \( B = (b_1, b_2, b_3) \) is given by:
\[
A \cdot B = a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3
\]
In our case, vector \( A \) is \( 6i + 4j - 2k \) and vector \( B \) is \( 5i - 6j - 3k \).
1. Identify the components of vector \( A \):
- \( a_1 = 6 \)
- \( a_2 = 4 \)
- \( a_3 = -2 \)
2. Identify the components of vector \( B \):
- \( b_1 = 5 \)
- \( b_2 = -6 \)
- \( b_3 = -3 \)
3. Apply the formula for the scalar product:
\[
A \cdot B = (6 \cdot 5) + (4 \cdot (-6)) + (-2 \cdot (-3))
\]
4. Calculate each term:
- \( 6 \cdot 5 = 30 \)
- \( 4 \cdot (-6) = -24 \)
- \( -2 \cdot (-3) = 6 \)
5. Add these results together:
\[
A \cdot B = 30 + (-24) + 6
\]
6. Simplify the expression:
\[
A \cdot B = 30 - 24 + 6
\]
\[
A \cdot B = 12
\]
Therefore, the scalar product (dot product) of vectors \( A \) and \( B \) is \( \boxed{12} \). This confirms that option C, which states 12, is the correct answer.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
