Verify that the scalar product defines an inner product on by establishing all…

MATHEMATICS · Subjective · CBSE · Mathematics · Vector Algebra · Vector Algebra

Problem

Verify that the scalar product defines an inner product on

\mathbb R^{3}

by establishing all required properties. (a) Prove that

\mathbf a\cdot\mathbf b=\mathbf b\cdot\mathbf a

for all

\mathbf a,\mathbf b\in\mathbb R^{3}

. (b) Prove that

(k\mathbf a)\cdot\mathbf b=k\,(\mathbf a\cdot\mathbf b)

for every scalar

k\in\mathbb R

. (c) Prove that

\mathbf a\cdot(\mathbf b+\mathbf c)=\mathbf a\cdot\mathbf b+\mathbf a\cdot\mathbf c

for all vectors

\mathbf a,\mathbf b,\mathbf c

. (d) Prove that

\mathbf a\cdot\mathbf a>0

for every non-zero vector

\mathbf a