The scalar product of 5i + j – 3k and 3i – 4j + 7k is:

Answer

To find the scalar product (or dot product) of the vectors \( \mathbf{A} = 5\mathbf{i} + \mathbf{j} - 3\mathbf{k} \) and \( \mathbf{B} = 3\mathbf{i} - 4\mathbf{j} + 7\mathbf{k} \), we can follow these steps:

### Step 1: Identify the components of the vectors
The first vector \( \mathbf{A} \) has components:
- \( a_1 = 5 \) (coefficient of \( \mathbf{i} \))
- \( a_2 = 1 \) (coefficient of \( \mathbf{j} \))
- \( a_3 = -3 \) (coefficient of \( \mathbf{k} \))

The second vector \( \mathbf{B} \) has components:
- \( b_1 = 3 \) (coefficient of \( \mathbf{i} \))
- \( b_2 = -4 \) (coefficient of \( \mathbf{j} \))
- \( b_3 = 7 \) (coefficient of \( \mathbf{k} \))

### Step 2: Apply the formula for the dot product
The formula for the dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \]

### Step 3: Substitute the values into the formula
Substituting the values we identified:
\[ \mathbf{A} \cdot \mathbf{B} = (5)(3) + (1)(-4) + (-3)(7) \]

### Step 4: Calculate each term
Calculating each term:
- First term: \( 5 \times 3 = 15 \)
- Second term: \( 1 \times -4 = -4 \)
- Third term: \( -3 \times 7 = -21 \)

### Step 5: Combine the results
Now, combine the results:
\[ \mathbf{A} \cdot \mathbf{B} = 15 - 4 - 21 \]

### Step 6: Simplify the expression
Calculating the final result:
\[ 15 - 4 = 11 \]
\[ 11 - 21 = -10 \]

Thus, the scalar product of the vectors \( \mathbf{A} \) and \( \mathbf{B} \) is:
\[ \mathbf{A} \cdot \mathbf{B} = -10 \]

### Final Answer:
The scalar product of \( 5\mathbf{i} + \mathbf{j} - 3\mathbf{k} \) and \( 3\mathbf{i} - 4\mathbf{j} + 7\mathbf{k} \) is \( -10 \).

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