By Walter Greiner
The sequence of texts on Classical Theoretical Physics relies at the hugely profitable sequence of classes given by way of Walter Greiner on the Johann Wolfgang Goethe college in Frankfurt am major, Germany. meant for complicated undergraduates and starting graduate scholars, the volumes within the sequence supply not just an entire survey of classical theoretical physics but in addition a huge variety of labored examples and difficulties to teach scholars in actual fact the way to follow the summary ideas to lifelike difficulties.
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If two row vectors (or column vectors) are equal or proportional to each other, then the determinant equals zero. 2. When we permute two neighboring rows, the determinant changes by a factor (−1). 1: Distance vector z = (x2 e1 + y2 e2 + z 2 e3 ) − (x1 e1 + y1 e2 + z 1 e3 ) y a r1 = a = r2 − r1 Solution P1 (x 1 ,y 1 ,z 1) Calculate the length of the vector a that represents the distance vector between the points r1 and r2 . 2: (x2 − x1 )2 + (y2 − y1 )2 + (z 2 − z 1 )2 . The distance vector between the points r1 and r2 .
17 is scalar-multiplied successively by the vectors ei × e j (i = j), where (ei × e j ) by deﬁnition points perpendicular to ei , hence the scalar products ei · (ei × e j ) vanish. 17 for Fi (i = 1, 2, 3) F1 = − F · (e2 × e3 ) , e1 · (e2 × e3 ) F2 = − F · (e3 × e1 ) , e2 · (e3 × e1 ) F3 = − F · (e1 × e2 ) . 19) e3 = (0, sin β, − cos β). 21) = 2 sin α cos α cos β. From there one obtains for the component F1 F1 = 1 2 Fy Fz tan β Fx − − cos α sin α sin α A z α y α β . 22) B F D C z F2 x F y x F1 D F3 A three-leg stand ﬁxed at a vertical wall.
This is the point-direction form of the equation of the plane. 4: The cosine theorem The cosine law of plane trigonometry is obtained by scalar multiplication of the equation c = a − b by itself: c · c = (a − b) · (a − b) = a 2 + b 2 − 2a · b = a + b − 2ab cos γ . 2 ⇒ 2 c2 = a 2 + b2 − 2ab cos γ . For γ = π/2 there results the theorem of Pythagoras. b γ c a The vectors a, b, and c characterize the sides of the triangle. 5: The theorem of Thales In order to prove the theorem of Thales1 we introduce the following vectors according to the sketch: −→ −→ M A = − M B = a, C ϑ −→ MC = b.