Scalar Product of Magnitude by Unit Vector Quantity

Theorem

Let $\mathbf a$ be a vector quantity.

Let $m$ be a scalar quantity.

Then:

$m \mathbf a = m \paren {\size {\mathbf a} \hat {\mathbf a} } = \paren {m \size {\mathbf a} } \hat {\mathbf a}$

where:

$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$

$\hat {\mathbf a}$ denotes the unit vector in the direction $\mathbf a$.

Proof

\(\ds \mathbf a\)

\(=\)

\(\ds \size {\mathbf a} \hat {\mathbf a}\)

Vector Quantity as Scalar Product of Unit Vector Quantity

\(\ds \leadsto \ \ \)

\(\ds m \mathbf a\)

\(=\)

\(\ds m \paren {\size {\mathbf a} \hat {\mathbf a} }\)

Then:

This needs considerable tedious hard slog to complete it. In particular: hard to prove something trivial To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

$\blacksquare$

Sources

• 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Addition and Subtraction of Vectors: $5$. Multiplication by a number