For the two vectors A and B in Fig. E1.39, find the scalar product A · B

Verified step by step guidance

1. Identify the magnitudes of vectors A and B from the diagram: A = 3.60 m and B = 2.40 m.
2. Determine the angle between the two vectors. From the diagram, the angle between A and B is 70° + 30° = 100°.
3. Recall the formula for the scalar product (dot product) of two vectors: A · B = |A| |B| cos(θ), where θ is the angle between the vectors.
4. Substitute the known values into the formula: A · B = (3.60 m) * (2.40 m) * cos(100°).
5. Calculate the cosine of 100° and multiply the values to find the scalar product A · B.

Key Concepts

Dot Product

The dot product, or scalar product, of two vectors A and B is a mathematical operation that results in a scalar. It is calculated as A · B = |A| |B| cos(θ), where |A| and |B| are the magnitudes of the vectors and θ is the angle between them. This operation is useful for determining the extent to which two vectors point in the same direction.

Vector Components

Vectors can be broken down into their components along the x and y axes. For a vector A at an angle θ, the components are given by Ax = |A| cos(θ) and Ay = |A| sin(θ). Understanding vector components is essential for calculating the dot product, as it allows for the direct multiplication of corresponding components of the vectors.

Angle Between Vectors

The angle between two vectors is crucial for calculating the dot product. In this case, the angle θ is the angle formed between the two vectors when placed tail-to-tail. It is important to accurately determine this angle, as it directly influences the cosine value used in the dot product formula, affecting the final scalar result.