By Percival Frost Joseph Wolstenholme

This Elibron Classics booklet is a facsimile reprint of a 1863 version by way of Macmillan and Co., Cambridge - London.

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Suppose two forces are applied to some object. Each of these would be represented by a force vector and the two forces acting together would yield an overall force acting on the which would be a force vector known as the resultant. Suppose ∑also ∑object n n the two vectors are a = k=1 ai ei and b = k=1 bi ei . Then the vector a involves a component in the ith direction, ai ei while the component in the ith direction of b is bi ei . Then it seems physically reasonable that the resultant vector should have a component in the ith direction equal to (ai + bi ) ei .

Thus the length of x equals x21 + x22 + x23 . When you multiply x by a scalar α, you get (αx1 , αx2 , αx3 ) and the length of this vector is deﬁned as √( √ ) 2 2 (αx1 ) + (αx2 ) + (αx3 ) 2 = |α| x21 + x22 + x23 . Thus the following holds. |αx| = |α| |x| . In other words, multiplication by a scalar magniﬁes or shrinks the length of the vector. What about the direction? You should convince yourself by drawing a picture that if α is negative, it causes the resulting vector to point in the opposite direction while if α > 0 it preserves the direction the vector points.

14 If z ∈ C there exists θ ∈ C such that θz = |z| and |θ| = 1. Proof: Let θ = 1 if z = 0 and otherwise, let θ = z . Recall that for z = x + iy, z = x − iy and |z| 2 zz = |z| . I will give a proof of this important inequality which depends only on the above list of properties of the inner product. It will be slightly diﬀerent than the earlier proof. 15 (Cauchy Schwarz)The following inequality holds for x and y ∈ Cn . |(x · y)| ≤ (x · x) 1/2 (y · y) 1/2 Equality holds in this inequality if and only if one vector is a multiple of the other.