(I) If vector A→ points along the negative x axis and vector B→ along the positive z axis, what is the direction of (a) A→ x B→ and (b) B→ x A→? (c) What is the magnitude of A→ x B→ and B→ x A→?

Verified step by step guidance

1. Identify the direction of each vector: Vector A→ points along the negative x-axis, and vector B→ points along the positive z-axis.
2. Use the right-hand rule to determine the direction of the cross product for each case. For A→ x B→, point your fingers in the direction of A→ (negative x-axis), then curl them towards B→ (positive z-axis). Your thumb will point in the direction of A→ x B→.
3. Apply the right-hand rule again for B→ x A→. Point your fingers in the direction of B→ (positive z-axis), then curl them towards A→ (negative x-axis). Your thumb will point in the direction of B→ x A→.
4. To find the magnitude of the cross product, use the formula |A→ x B→| = |A| |B| sin(θ), where θ is the angle between vector A→ and vector B→. Since the vectors are perpendicular, sin(θ) = 1.
5. Since the magnitude of A→ x B→ and B→ x A→ are calculated using the same formula and the vectors are of equal magnitude and perpendicular, the magnitudes of A→ x B→ and B→ x A→ will be the same.

Key Concepts

Vector Cross Product

The vector cross product is a mathematical operation that takes two vectors and produces a third vector that is perpendicular to the plane formed by the original vectors. The direction of the resulting vector is determined by the right-hand rule, which states that if you curl the fingers of your right hand from the first vector to the second, your thumb points in the direction of the cross product.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the cross product of two vectors. To apply it, extend your right hand with your fingers pointing in the direction of the first vector and curl them towards the second vector. Your thumb will then point in the direction of the resulting vector from the cross product.

Magnitude of Cross Product

The magnitude of the cross product of two vectors is calculated using the formula |A x B| = |A| |B| sin(θ), where |A| and |B| are the magnitudes of the vectors and θ is the angle between them. In the case of perpendicular vectors, the sine of 90 degrees is 1, making the magnitude equal to the product of the magnitudes of the two vectors.