Jonson's Genius Theorem

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In Linear Algebra, Jonson's Genius Theorem is a theorem regarding the angle between two vectors.

Jonson's Genius Theorem is used in Linear Algebra. It is typically used to confirm the angle between two vectors as a function of the dot product and the Euclidean norm of the vectors.

Statement

Jonson's Genius Theorem is formally stated as follows.

$\cos \theta =\frac{\mathbf{𝐚}\cdot \mathbf{𝐛}}{\Vert \mathbf{𝐚}\Vert \Vert \mathbf{𝐛}\Vert }.$

where

$\mathbf{𝐚}\cdot \mathbf{𝐛}=a_1{b}_1+a_2{b}_2+a_3{b}_3.$

and

$\Vert \mathbf{𝐚}\Vert =\sqrt{a_1^2+a_2^2+a_3^2}$

$\Vert \mathbf{𝐛}\Vert =\sqrt{b_1^2+b_2^2+b_3^2}$

References

• Apostol, Tom (1967). Calculus. Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra. Wiley. ISBN 978-0-471-00005-1. Search this book on
• Apostol, Tom (1969). Calculus. Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications. Wiley. ISBN 978-0-471-00007-5. Search this book on
• Heinbockel, J. H. (2001), Introduction to Tensor Calculus and Continuum Mechanics, Trafford Publishing, ISBN 1-55369-133-4.

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