Dot and cross products of two vectors in 2D or 3D, their magnitudes, the angle between them and the projection of one onto the other, with a drawing.
업데이트 검증한 예제: 5
a · b = 32, so the vectors are 12.9332° apart; the parallelogram they span has area 7.34847.
Equivalent to arccos(a·b / |a||b|), but stays accurate when the vectors are nearly parallel.
The dot product multiplies matching components and adds them, a · b = a₁b₁ + a₂b₂ + a₃b₃. The cross product of two 3D vectors is a vector perpendicular to both, and its length equals the area of the parallelogram they span. The angle between the vectors is computed as θ = atan2(|a × b|, a · b), which equals arccos(a · b / |a||b|) but stays accurate for nearly parallel vectors, and the projection of a onto b is (a · b / |b|²) b.
Physics (work and torque), 3D graphics and linear algebra are the usual uses. The default vectors a = (1, 2, 3) and b = (4, 5, 6) give a · b = 32, a × b = (−3, 6, −3) and an angle of about 12.93°.
A dot product of 0 means the vectors are perpendicular; a zero cross product means they are parallel or one is zero. In 2D mode both vectors lie in the xy-plane, so their cross product points along z: (3, 4) × (4, −3) = (0, 0, −25).
검증 출처: Hand calculation; Python decimal √14, √54 and atan2(√54, 32) in degrees (hp.py)
검증 출처: 3·4 + 4·(−3) = 0; 3·(−3) − 4·4 = −25
검증 출처: b = 2a, so a × b = 0 and θ = 0
검증 출처: a·b/|b| = 8/4; angle atan2(3, 2) = 56.3099324740202…° (Python math.degrees)
Multiply corresponding components and add the results: a · b = a₁b₁ + a₂b₂ + a₃b₃. For (1, 2, 3) · (4, 5, 6) that is 4 + 10 + 18 = 32. The dot product also equals |a||b| cos θ, so it is positive when the angle is under 90°, 0 at exactly 90° and negative beyond it: (1, 0) · (−2, 0) = −2.
For a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For (1, 2, 3) × (4, 5, 6) that gives (12 − 15, 12 − 6, 5 − 8) = (−3, 6, −3). The result is perpendicular to both vectors, follows the right-hand rule, and changes sign if the order is swapped: b × a = (3, −6, 3).
Use cos θ = (a · b)/(|a||b|). For (1, 2, 3) and (4, 5, 6), cos θ = 32/(√14 × √77) ≈ 0.974632, so θ ≈ 12.93°. Near 0° or 180° the arccos form loses accuracy, so the calculator uses the equivalent θ = atan2(|a × b|, a · b), the form William Kahan recommends in his notes on floating-point roundoff.
The two vectors are perpendicular (orthogonal), provided neither is the zero vector. (3, 4) · (4, −3) = 12 − 12 = 0, so those vectors meet at exactly 90°. In physics the same test shows that a force at right angles to the motion does no work, since work is the dot product of force and displacement.
The scalar projection of a onto b is the signed length of a along b, a · b / |b|; the vector projection is that length times the unit vector of b, (a · b / |b|²) b. Projecting (2, 3) onto (4, 0) gives a scalar projection of 8/4 = 2 and a vector projection of (2, 0). A negative scalar projection means a points partly against b.
정확도는 입력값과 계산 방법의 가정에 따라 달라집니다. 십진 연산은 유효숫자 50자리를 사용하지만, 추정값·수치해석 방법·원본 데이터의 정밀도는 더 낮을 수 있습니다. 표시값을 반올림해도 이러한 한계는 사라지지 않습니다. 독립적인 출처의 풀이와 대조한 계산 예시: 5. 예를 들어 “a = (1, 2, 3), b = (4, 5, 6)”은 Hand calculation; Python decimal √14, √54 and atan2(√54, 32) in degrees (hp.py)와 대조해 확인합니다.
Wolfram MathWorld — Dot Product; Wolfram MathWorld — Cross Product; W. Kahan, How futile are mindless assessments of roundoff in floating-point computation? §12 — angles via atan2 rather than arccos.
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