Find the scalar product of the vectors A = 7i - 2j + k and B = 3i + 4j - 5k.

Find the scalar product of the vectors A = 7i - 2j + k and B = 3i + 4j - 5k.

Step 1: Identify the components of vector A. A = 7i - 2j + k means A has components: A_x = 7, A_y = -2, A_z = 1.
Step 2: Identify the components of vector B. B = 3i + 4j - 5k means B has components: B_x = 3, B_y = 4, B_z = -5.
Step 3: Calculate the product of the x-components: A_x * B_x = 7 * 3 = 21.
Step 4: Calculate the product of the y-components: A_y * B_y = -2 * 4 = -8.
Step 5: Calculate the product of the z-components: A_z * B_z = 1 * -5 = -5.
Step 6: Add all the products together: 21 + (-8) + (-5) = 21 - 8 - 5.
Step 7: Simplify the addition: 21 - 8 = 13, then 13 - 5 = 8.
Step 8: The scalar product A · B is 8.

Scalar Product – The scalar product (or dot product) of two vectors is calculated by multiplying their corresponding components and summing the results.
Vector Components – Understanding how to break down vectors into their i, j, and k components is essential for performing operations like the scalar product.