In this example, we explore the calculation of the dot product and the angle between two vectors, a and b, represented in unit vector components. The dot product, denoted as a · b, can be calculated using the formula that involves the components of the vectors. For vectors expressed in terms of their unit components, the dot product is computed by pairing the corresponding components and summing the products. Specifically, if a is given as 7.2i - 3.9j and b as 2.1i + 4.8j, the dot product is calculated as follows:

a · b = (7.2)(2.1) + (-3.9)(4.8)

Calculating this gives:

a · b = 15.12 - 18.72 = -3.6

This result indicates that the dot product is -3.6, which can be interpreted as a measure of how aligned the two vectors are; a negative value suggests they point in somewhat opposite directions.

Next, we determine the angle θ between the two vectors. The relationship between the dot product and the angle is given by the equation:

a · b = |a| |b| cos(θ)

To find θ, we first need to calculate the magnitudes of both vectors. The magnitude of a vector can be found using the Pythagorean theorem:

|a| = √(7.2² + (-3.9)²) = √(51.84 + 15.21) = √67.05 ≈ 8.2

|b| = √(2.1² + 4.8²) = √(4.41 + 23.04) = √27.45 ≈ 5.2

Now, substituting the values into the dot product equation:

-3.6 = (8.2)(5.2) cos(θ)

Solving for cos(θ) gives:

cos(θ) = -3.6 / (8.2 * 5.2) ≈ -0.08

To find the angle θ, we take the inverse cosine:

θ = cos^-1(-0.08) ≈ 94.6°

This angle indicates that the vectors are nearly perpendicular, as expected from the negative cosine value. A sketch of the vectors confirms their orientation, with vector a having a positive x-component and a negative y-component, while vector b has both positive components, resulting in an angle slightly greater than 90 degrees.