By Douady R., Douady A.

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**Example text**

In other words, the angle α is the angle ˆ1 and a. Similarly, the angles β and γ are, by deﬁnition, the between e ˆ3 , respectively. Then ˆ2 and e angles between a and the unit vectors e cos α = ˆ2 · a ˆ1 · a e a1 a2 e , cos β = , = = ˆ1 a ˆ2 a e a e a ˆ3 · a a3 e = . cos γ = ˆ3 a e a 26 11. 9. Left: Direction angles of a vector are deﬁned as the angles between the vector and three coordinates axes. Each angle ranges between 0 and π and is counted from the corresponding positive coordinate semiaxis toward the vector.

Two lines that are not intersecting and not parallel are called skew lines. To determine the distance between skew lines L1 and L2 , consider any two points A and B on L1 and any two points C and P on L2 . Deﬁne the vectors b = AB and c = CP that are parallel to lines L1 and L2 , respectively. Since the lines are not parallel, the cross product b × c does not vanish. The lines L1 and L2 lie in the parallel planes perpendicular to b × c (by the geometrical properties of the cross product, b × c is perpendicular to b and c).

C1 c2 c3 This provides a convenient way to calculate the numerical value of the triple product. If two rows of a matrix are swapped, then its determinant changes sign. Therefore, a · (b × c) = −b · (a × c) = −c · (b × a) . This means, in particular, that the absolute value of the triple product is independent of the order of the vectors in the triple product. 75. 12. Left: Geometrical interpretation of the triple product as the volume of the parallelepiped whose adjacent sides are the vectors in the product: h = a cos θ, A = b × c , V = hA = a b × c cos θ = a · (b × c).