Show that î x ĵ = k̂ , î x k̂ = - ĵ, and ĵ x k̂ = î.

검증된 단계별 안내

Understand the problem: The task is to verify the cross products of unit vectors in a Cartesian coordinate system. The unit vectors î, ĵ, and k̂ represent the x, y, and z directions, respectively. The cross product of two vectors results in a vector perpendicular to both, following the right-hand rule.

Recall the right-hand rule: To determine the direction of the cross product, point your right-hand fingers in the direction of the first vector (e.g., î), curl them toward the second vector (e.g., ĵ), and your thumb will point in the direction of the resulting vector (e.g., k̂).

Verify î x ĵ = k̂: Using the right-hand rule, point your fingers in the direction of î (x-axis) and curl them toward ĵ (y-axis). Your thumb points in the direction of k̂ (z-axis), confirming that î x ĵ = k̂.

Verify î x k̂ = -ĵ: Point your fingers in the direction of î (x-axis) and curl them toward k̂ (z-axis). Your thumb points in the negative y-direction, confirming that î x k̂ = -ĵ.

Verify ĵ x k̂ = î: Point your fingers in the direction of ĵ (y-axis) and curl them toward k̂ (z-axis). Your thumb points in the direction of î (x-axis), confirming that ĵ x k̂ = î.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

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

Vector Cross Product

The vector cross product is a binary operation on two vectors in three-dimensional space, resulting in a third vector that is perpendicular to the plane formed by the original vectors. The magnitude of the cross product is given by the product of the magnitudes of the two vectors and the sine of the angle between them. This operation is denoted by the symbol '×' and is essential for determining torque, angular momentum, and other vector quantities.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to indicate direction. In three-dimensional space, the standard unit vectors are î, ĵ, and k̂, which represent the x, y, and z axes, respectively. They are fundamental in vector analysis, allowing for the representation of any vector as a combination of these unit vectors, simplifying calculations involving direction and magnitude.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the resultant vector in a cross product. By aligning the fingers of the right hand in the direction of the first vector and curling them towards the second vector, the thumb points in the direction of the cross product. This rule is crucial for visualizing and understanding the orientation of vectors in three-dimensional space.