Show that if v1, v2, and v3 are mutually orthogonal nonzerovectors in 3-space, and if a

Chapter 11, Problem 45

(choose chapter or problem)

Show that if \(v_{1}, v_{2}\), and \(v_{3}\) are mutually orthogonal nonzero vectors in 3 -space, and if a vector v in 3 -space is expressed as

\(v=c_{1} v_{1}+c_{2} v_{2}+c_{3} v_{3}\)

then the scalars \(c_{1}, c_{2}\), and \(c_{3}\) are given by the formulas

\(c_{i}=\left(v \cdot v_{i}\right) /\left\|v_{i}\right\|^{2}, i=1,2,3\)

Equation Transcription:

Text Transcription:

v_1, v_2

v_3

v=c_1+v_1+c_2v_2+c_3+v_3

c_1, c_2

c_3

c_1=(v times v_1)/||v_1||^2, i=1,2,3