Evaluate the cross products and A ✕ B and C ✕ D.

검증된 단계별 안내

Step 1: Understand the cross product formula. The magnitude of the cross product of two vectors A and B is given by |A ✕ B| = |A| |B| sin(θ), where |A| and |B| are the magnitudes of the vectors and θ is the angle between them. The direction of the cross product is perpendicular to the plane containing A and B, determined by the right-hand rule.

Step 2: Analyze the image. The vectors F and E are shown with magnitudes 4 and 3, respectively, and the angle between them is 30°. This information will be used to calculate the magnitude of the cross product.

Step 3: Calculate the magnitude of F ✕ E. Using the formula |F ✕ E| = |F| |E| sin(θ), substitute the values: |F| = 4, |E| = 3, and θ = 30°. The sine of 30° is 0.5.

Step 4: Determine the direction of F ✕ E. Use the right-hand rule: point your fingers in the direction of F, curl them toward E, and your thumb will point in the direction of the cross product vector.

Step 5: Repeat the process for vectors C and D if their magnitudes and angle are provided. Apply the same formula and right-hand rule to evaluate C ✕ D.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Cross Product

The cross product of two vectors results in a third vector that is perpendicular to the plane formed by the original vectors. It is calculated using the formula A × B = |A||B|sin(θ)n, where θ is the angle between the vectors and n is the unit vector perpendicular to the plane. The magnitude of the cross product represents the area of the parallelogram formed by the two vectors.

Vector Magnitude

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem for two-dimensional vectors. For a vector represented as A = (Ax, Ay), the magnitude is given by |A| = √(Ax² + Ay²). In the context of the question, the magnitudes of vectors E and F are 3 and 4, respectively, which are essential for calculating their cross product.

Angle Between Vectors

The angle between two vectors is crucial for determining the sine component in the cross product calculation. In this case, the angle is given as 30°, which affects the magnitude of the resulting vector from the cross product. Understanding how to find and use this angle is key to solving problems involving vector operations.