Like Vector Quantities are Multiples of Each Other

Theorem

Let $\mathbf a$ and $\mathbf b$ be like vector quantities.

Then:

$\mathbf a = \dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$

where:

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

$\dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$ denotes the scalar product of $\mathbf b$ by $\dfrac {\size {\mathbf a} } {\size {\mathbf b} }$.

Proof

By the definition of like vector quantities:

$\mathbf a$ and $\mathbf b$ are like vector quantities if and only if they have the same direction.

By definition of unit vector:

$\dfrac {\mathbf a} {\size {\mathbf a} } = \dfrac {\mathbf b} {\size {\mathbf b} }$

as both are in the same direction, and both have length $1$.

By definition of scalar division:

$\dfrac 1 {\size {\mathbf a} } \mathbf a = \dfrac 1 {\size {\mathbf b} } \mathbf b$

Hence, multiplying by $\size {\mathbf a}$:

$\mathbf a = \dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$

$\blacksquare$

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