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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.
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