2.9
ベクトル積（外積）
2 つのベクトルのベクトル乗算は外積を生成します。その大きさは、個々のベクトルと両方のベクトルの間の角度の正弦と両方の個々のベクトルに垂直な方向の積に等しくなります。 特定の平面に垂直な方向は常に2つあり、どちらの方向にも1つずつ存在します。外積の方向は右手の法則によって決まります。
2 つのベクト…
2 つのベクトルの交差積またはベクトル積は、それらの大きさと角度の正弦の積です。ベクトルの平面に対して垂直に向けられます。
幾何学的には、ベクトルが広がる平行四辺形の面積です。
右手の法則を使用すると、右手の人差し指と中指がそれぞれ 1 番目と 2 番目のベクトルを指している場合、親指の指はそれらの外積を指します。
2 番目のベクトルと 1 番目のベクトルの外積は反対方向です。したがって、クロス積の順序は重要です。
ドアの回転の測定値は、ヒンジと力が加えられた力の印加点との間の半径方向の距離の外積であるトルクから得られます。
半径方向の距離の方向に沿って、または反対方向に力が作用すると、ドアは回転しません。
軸の単位ベクトルの場合、軸自体との外積は 0 です。しかし、もう 1 つでは、巡回順の 3 番目の 1 つです。
ベクトル A と B の外積の成分は、AyBz z から AzBy を引いたもの、AzBx から AxBz を引いたもの、AxBy を引いたもの AyBx です。
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Q1: What is the cross product and how is it calculated geometrically?
The cross product of two vectors equals the product of their magnitudes and the sine of the angle between them, directed perpendicular to their plane. Geometrically, it represents the area of the parallelogram spanned by the two vectors. The magnitude depends on both vector sizes and their angular relationship, while direction follows the right-hand rule.
Q2: How does the right-hand rule determine cross product direction?
Point your right hand's index and middle fingers along the first and second vectors respectively. Your thumb then points in the cross product direction. If you reverse the vector order, the thumb points opposite, showing that cross products are anti-commutative. This rule resolves the ambiguity of two perpendicular directions.
Q3: Why does the order of vectors matter in cross product calculations?
The cross product is anti-commutative: reversing vector order reverses the result's direction. The cross product of vector B with vector A points opposite to the cross product of A with B. This directional reversal means A × B ≠ B × A, making order critical for correct physical interpretation.
Q4: What are the component formulas for calculating a cross product?
For vectors A and B, the cross product components are: (AyBz − AzBy), (AzBx − AxBz), and (AxBy − AyBx). These formulas use vector components in the Cartesian coordinate system to compute each component of the resulting cross product vector systematically.
Q5: When is the cross product of two vectors equal to zero?
The cross product equals zero when vectors are parallel or antiparallel, including when a vector is crossed with itself. Since the sine of 0° and 180° is zero, any vector pair aligned in the same or opposite direction produces zero cross product, regardless of magnitude.
Q6: How does torque relate to the cross product in physics?
Torque is the cross product of the radial distance vector from a hinge with the applied force vector. A door rotates only when force acts perpendicular to the radial distance; force along or opposite the radial direction produces zero torque. This demonstrates how cross product direction determines rotational effect.
Q7: What is the cross product of unit vectors in cyclic order?
Unit vectors of perpendicular axes follow a cyclic pattern: each unit vector crossed with itself yields zero, but crossed with another unit vector yields the third one in cyclic order. This property simplifies calculations and reflects the fundamental orthogonal structure of three-dimensional space.