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

To find the scalar product (also known as the dot product) of the vectors \(5i + j - 3k\) and \(3i - 4j + 7k\), we can use the formula:

Dot Product Formula

The dot product of two vectors \(A = ai + bj + ck\) and \(B = xi + yj + zk\) is calculated as:

A · B = ax + by + cz

Applying the Formula

For the vectors given:

• Vector A: \(5i + j - 3k\) (where \(a = 5\), \(b = 1\), \(c = -3\))
• Vector B: \(3i - 4j + 7k\) (where \(x = 3\), \(y = -4\), \(z = 7\))

Calculating Each Component

Now, we can substitute the values into the dot product formula:

• First component: \(5 \times 3 = 15\)
• Second component: \(1 \times -4 = -4\)
• Third component: \(-3 \times 7 = -21\)

Summing the Results

Now, add these results together:

15 - 4 - 21 = -10

Final Answer

The scalar product of the vectors \(5i + j - 3k\) and \(3i - 4j + 7k\) is -10.