Find the vector product A x B (expressed in unit vectors) of the two vectors given in Exercise 1.38. What is the magnitude of the vector product? Given two vectors A = 4.00 i + 7.00j and B = 5.00 i − 2.00

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1. Understand the vector product (also known as the cross product) of two vectors A and B. The cross product is a vector that is perpendicular to both A and B and is given by the formula: A x B = (A_y * B_z - A_z * B_y) i + (A_z * B_x - A_x * B_z) j + (A_x * B_y - A_y * B_x) k.
2. Identify the components of vectors A and B. For vector A = 4.00 i + 7.00 j, the components are A_x = 4.00, A_y = 7.00, and A_z = 0 (since there is no k component). For vector B = 5.00 i - 2.00 j, the components are B_x = 5.00, B_y = -2.00, and B_z = 0.
3. Substitute the components of vectors A and B into the cross product formula. Since both vectors have no k component, the formula simplifies to: A x B = (7.00 * 0 - 0 * -2.00) i + (0 * 5.00 - 4.00 * 0) j + (4.00 * -2.00 - 7.00 * 5.00) k.
4. Calculate each component of the resulting vector from the cross product. The i and j components will be zero due to the absence of k components in A and B, and the k component will be calculated using the values: (4.00 * -2.00 - 7.00 * 5.00).
5. Determine the magnitude of the vector product. The magnitude of a vector V = V_x i + V_y j + V_z k is given by the formula: |V| = sqrt(V_x^2 + V_y^2 + V_z^2). Since the i and j components are zero, the magnitude will be |V| = sqrt((4.00 * -2.00 - 7.00 * 5.00)^2).

Key Concepts

Vector Product (Cross Product)

The vector product, or cross product, of two vectors results in a third vector that is perpendicular to the plane containing the original vectors. It is calculated using the determinant of a matrix formed by unit vectors and the components of the given vectors. The direction follows the right-hand rule, and the magnitude is given by |A||B|sin(θ), where θ is the angle between A and B.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to indicate direction in space. In Cartesian coordinates, the standard unit vectors are i, j, and k, representing the x, y, and z axes, respectively. They are essential in expressing vectors in component form, allowing for operations like addition, subtraction, and cross product to be performed algebraically.

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size, calculated using the square root of the sum of the squares of its components. For a vector resulting from a cross product, the magnitude can be found using the formula |A x B| = |A||B|sin(θ), which represents the area of the parallelogram formed by the original vectors. This is crucial for understanding the scale of the resulting vector.