Linear Algebra

42 Chapter One. Linear SystemsThe Cauchy-Schwartz inequality assures us that the next definition makessense because the fraction has absolute value less than or equal to one.2.7 Definition The angle between two nonzero vectors ⃗u, ⃗v ∈ R n isθ = arccos(⃗u ⃗v‖⃗u ‖ ‖⃗v ‖ )(the angle between the zero vector and any other vector is defined to be a rightangle).Thus vectors from R n are orthogonal if and only if their dot product is zero.2.8 Example These vectors are orthogonal.( ( )1 1= 0−1)1The arrows are shown away from canonical position but nevertheless the vectorsare orthogonal.2.9 Example The R 3 angle formula given at the start of this subsection is aspecial case of the definition. Between these two( ) 032the angle is( ) 110(1)(0) + (1)(3) + (0)(2)arccos( √12 + 1 2 + 0 2√ 0 2 + 3 2 + 2 ) = arccos( 3√ √ )2 2 13approximately 0.94 radians. Notice that these vectors are not orthogonal. Althoughthe yz-plane may appear to be perpendicular to the xy-plane, in factthe two planes are that way only in the weak sense that there are vectors in eachorthogonal to all vectors in the other. Not every vector in each is orthogonal toall vectors in the other.Exerciseš 2.10 Find the length of each vector.