Magnitude of Vector Quantity in terms of Components

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Theorem

Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space.

Let $\mathbf r$ be expressed in terms of its components:

$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$

where $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively.


Then:

$\size {\mathbf r} = \sqrt {x^2 + y^2 + z^2}$

where $\size {\mathbf r}$ denotes the magnitude of $\mathbf r$.


Proof

Let the initial point of $\mathbf r$ be $\tuple {x_1, y_1, z_1}$.

Let the terminal point of $\mathbf r$ be $\tuple {x_2, y_2, z_2}$.

Thus, by definition of the components of $\mathbf r$, the magnitude of $\mathbf r$ equals the distance between $\tuple {x_1, y_1, z_1}$ and $\tuple {x_2, y_2, z_2}$.

The result follows from Distance Formula in 3 Dimensions.

$\blacksquare$


Sources

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