Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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.
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
