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I'm reading the book 3d Math Primer for Graphics and Game Development 1st edition by Fletcher Dunn and Ian Parberry. On page 61 there's this:

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How does it derive $ v_{||} $ there? It doesn't follow from what preceded this point in the book, unless I'm missing something. Also I'm not sure, is $n$ supposed to be unit vector here? This should probably be in the errata but I didn't find anything. Anyway, how did he deduce this equation? Thanks in advance.

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Since v∥ is parallel to n, it can be expressed as some multiple of n. The multiple is the ratio of the length of v∥ to the length of n. The n vector doesn’t have to be normalized, but if it is, then |n| is 1, so the equation becomes v∥ = n * |v∥|.

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Since we have:

$$\pmb{v}_{\parallel} \parallel \pmb{n},$$

then:

$$\pmb{v}_{\parallel} = k\pmb{n}$$

We want that the length of $\pmb{v}_{\parallel}$ is $\|\pmb{v}_{\parallel}\|$, then:

$$\|\pmb{v}_{\parallel}\| =|k|\|\pmb{n}\| \implies |k| = \frac{\|\pmb{v}_{\parallel}\|}{\|\pmb{n}\|}$$

Then the sign of $k$ is simply chosen to match the direction of $\pmb{v}$ (if the angle between $\pmb{v}$ and $\pmb{n}$ is more than 90 degrees, the sign is negative, otherwise it is positive).

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