By Kleppner D., Kolenkow R.

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To illustrate this, here is a vector A drawn in two diﬀerent coordinate systems. In the first case, y A A = (A, 0) x (x, y system), while in the second x′ A Az = Bz . A = (0, −A) (x , y system). All vector operations can be written as equations for components. For instance, multiplication by a scalar is written cA = (cA x , cAy , cAz ). y′ The law for vector addition is A + B = (A x + Bx , Ay + By , Az + Bz ). By writing A and B as the sums of vectors along each of the coordinate axes, you can verify that A · B = A x Bx + Ay By + Az Bz .

5 Perpendicular vectors Show that if |A − B| = |A + B|, then A and B are perpendicular. 6 Diagonals of a parallelogram Show that the diagonals of an equilateral parallelogram are perpendicular. 7 Law of sines* Prove the law of sines using the cross product. It should only take a couple of lines. 8 Vector proof of a trigonometric identity Let aˆ and bˆ be unit vectors in the x−y plane making angles θ and φ with the x axis, respectively. Show that aˆ = cos θˆi + sin θˆj, bˆ = cos φˆi + sin φˆj, and using vector algebra prove that cos(θ − φ) = cos θ cos φ + sin θ sin φ.

However, by recasting the form of ΔT , both of these problems can be solved. The trick is to write ΔT as a power series in the small parameter x = l/L. We have ⎛ ⎜⎜⎜ g 1 − ΔT = T − T 0 = 2π ⎜⎜⎜⎜⎝ L 1 + l/L ⎞ ⎛ ⎟⎟ ⎜⎜⎜ 1 ⎜ = T 0 ⎜⎝ − 1⎟⎟⎟⎠ . 1+x ⎞ g ⎟⎟⎟⎟⎟ ⎟ L ⎟⎠ (1) NOTES 37 Next, we make use of the following identity, which will be derived in the following section: 1 3 1 1 3 = 1 − x + x2 − x + ··· 1+x 2 8 16 (2) This expansion is valid provided x < 1. Inserting this in Eq. (1) gives 1 3 1 3 ΔT = T 0 − x + x2 − x + ··· .