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14) Vectors a and b are perpendicular if and only if a· b = 0. If vectors a' and b' are parallel to non-zero perpendicular vectors a b respectively, then they are also perpendicular. 1lled. perpendicular. 14) that if a vector b is perpendic­ ular to each of the vectors au a2, , ak, then it is also perpendicular -and • • • I. e. + tkak, where t11 t2, ,tk are numeral coefficients. 8) implies that the cosine of the angle of the vector a = [a11 a2, ,an] =fa 0 with the versor lli is equal to ai/lal.

By the e(b,c)+e(a,c)+e(a,b) triangle inequality, the (the radicand perimeter is of the non-negative. Furthermore, it is independent of the order of the points a,b,c and depends only on their mutual distances. _bi)(lal-lbl +la-bl)(-lal+lb! 2) IL1(a,b,c)I = tlallblsinO. The number [al [bjsinO is called the area of the parallelogram spanned on the vectors a and b. Its intuitive sense is clear. EXERCISES I. , Qk be vectors of the space On and a1, a , • • • , ak numbers. 2 Show that if the vector a = a1 a1 + a2a2 + .

An isometry f of the space On onto itself is a translation if and only if there exists a number M such that e(x, f(x)) < M for each xeOn. 2) LEMMA. If