To translate a vector by 10 unit in the X direction, why do we have to use a matrix?
We can just add 10 to the mat[0][0], and we got the same result too.
Answer
Yes, you can add a vector in the case of translation. The reason to use a matrix boils down to having a uniform way to handle different combined transformations.
For example, rotation is usually done using a matrix (check @MickLH comment for other ways to deal with rotations), so in order to deal with multiple transformations (rotation/translation/scaling/projection...etc) in a uniform way, you need to encode them in a matrix.
Well, more technically speaking; a transformation is mapping a point/vector to another point/vector.
p` = T(p);
where p` is the transformed point and T(p) is the transformation function.
Given that we don't use a matrix we need to do this to combine multiple transformations:
p1= T(p);
pfinal = M(p1);
Not only can a matrix combine multiple types of transformations into a single matrix (e.g. affine, linear, projective).
Using a matrix gives us the opportunity to combine chains of transformations and then batch multiply them. This saves us a ton of cycles usually by the GPU (thanks to @ChristianRau for pointing it out).
Tfinal = T * R * P; // translaterotateproject
pfinal = Tfinal*p;
It's also good to point out that GPUs and even some CPUs are optimized for vector operations; CPUs using SIMD and GPUs being data driven parallel processors by design, so using matrices fits perfectly with hardware acceleration (actually, GPUs were designed to fit matrix/vector operations).
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