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I am reading Miroslav Lovric's book: Vector Calculus ... and am currently focused n Section 1.3: The Dot Product ...

I need help with an apparently simple matter involving Theorem 1.6 and the section on the orthogonal vector projection and the scalar projection ...My question is as follows:

It seems to me that \(\displaystyle a_v\) in Lovric's Theorem 1.6 should be equal to the scalar projection, \(\displaystyle \| \text{ proj}_v a \|\) ... BUT ... they are not given as equal ...'

... indeed ...

\(\displaystyle a_v = a \cdot v / \| v \|^2\)

while

\(\displaystyle \| \text{ proj}_v a \| = a \cdot v / \| v \|\)BUT ... ... geometrically they seem to be referring to the same quantity ... what then accounts for the difference in the formulae ...?Hope someone can help ...

Peter========================================================================================

The above post refers to Theorem 1.6 so I am providing the text of the same ... as follows:

View attachment 8677

View attachment 8678

The above post also refers to the text on the orthogonal vector projection and the scalar projection ... so I am providing the text of the same ... as follows:

View attachment 8679

View attachment 8680

Hope someone can clarify the issue in the above post ...

Peter

I need help with an apparently simple matter involving Theorem 1.6 and the section on the orthogonal vector projection and the scalar projection ...My question is as follows:

It seems to me that \(\displaystyle a_v\) in Lovric's Theorem 1.6 should be equal to the scalar projection, \(\displaystyle \| \text{ proj}_v a \|\) ... BUT ... they are not given as equal ...'

... indeed ...

\(\displaystyle a_v = a \cdot v / \| v \|^2\)

while

\(\displaystyle \| \text{ proj}_v a \| = a \cdot v / \| v \|\)BUT ... ... geometrically they seem to be referring to the same quantity ... what then accounts for the difference in the formulae ...?Hope someone can help ...

Peter========================================================================================

The above post refers to Theorem 1.6 so I am providing the text of the same ... as follows:

View attachment 8677

View attachment 8678

The above post also refers to the text on the orthogonal vector projection and the scalar projection ... so I am providing the text of the same ... as follows:

View attachment 8679

View attachment 8680

Hope someone can clarify the issue in the above post ...

Peter