Triangle Inequality The Triangle Inequality for
(a) Give a geometric interpretation of the Triangle Inequality.
(b) Use the Cauchy-Schwarz Inequality from Exercise 61 to prove the Triangle Inequality. [Hint: Use the fact that
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Calculus, Early Transcendentals
- Proof Complete the proof of the cancellation property of vector addition by justifying each step. Prove that if u, v, and w are vectors in a vector space V such that u+w=v+w, then u=v. u+w=v+wu+w+(w)=v+w+(w)a._u+(w+(w))=v+(w+(w))b._u+0=v+0c._ u=vd.arrow_forwardVerifying Inequalities In Exercises 53-64, verify a the Cauchy-Schwarz Inequality and b the triangle inequality for given vectors and inner products. u=(1,1),v=(1,1),u,v=uvarrow_forwardTrue or False? In Exercises 55 and 56, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a A set S of vectors in an inner product space V is orthonormal when every vector is a unit vector and each pair of vectors is orthogonal. b If a set of nonzero vectors S in an inner product space V is orthogonal, then S is linearly independent.arrow_forward
- Verifying the Cauchy-Schwarz Inequality In Exercises 35-38, verify the Cauchy-Schwarz inequality for the vectors. u=(6,8), v=(3,2).arrow_forwardCalculusIn Exercises 29 and 30, a find the inner product, b determine whether the vectors are orthogonal, and c verify the Cauchy-Schwarz Inequality for the vectors. f(x)=x,g(x)=1x2+1,f,g=11f(x)g(x)dxarrow_forwardProofProve in full detail that M2,2, with the standard operations, is a vector space.arrow_forward
- Verifying InequalitiesIn Exercises 53-64, verify a the Cauchy-Schwarz Inequality and b the triangle inequality for given vectors and inner products. Calculusf(x)=sinx, g(x)=cosx, f,g=0/4f(x)g(x)dxarrow_forwardProof Let u and v be a nonzero vectors in an inner product space V. Prove that uprojvu is orthogonal to v.arrow_forward
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