# Why translation uses multiplication and not addition? [closed]

To translate an object from (x, y, z) to (a + x, b + y, c + z) it's done using:

OK, that's cool, but why would we multiply in the first place when we can add the two matrices directly as:

addition is faster and more effective, plus it's straightforward?

## closed as primarily opinion-based by sam hocevar, MrCranky, Anko, bummzack, Maik SemderJan 17 '14 at 6:52

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• Isn't this more of a math question than a game development question? – Anko Jan 16 '14 at 10:08
• @Anko This is more about the implementation and applied part in graphics than the mathematical part. – GAX Jan 16 '14 at 10:22
• This question is mostly opinion-based. Addition is indeed more effective and straightforward, but what makes you think that it isn’t actually used? I add vectors together all the time. – sam hocevar Jan 16 '14 at 10:47
• Voting to close, as the premise of the question is wrong, since it assumes that translation is always done using multiplication, which is simply not true. It's a matter of opinion, personal preference and circumstance as to which one to use. No-one is forcing you to use full 4x4 matrix operations to do a simple translation. – MrCranky Jan 16 '14 at 12:22

because it allows to combine rotation and translation into a single matrix. Multiplying those matrices together gives use the concatenated operation. We can rotation around an arbitrary point P by first translating to -P then rotating and then translating by P.

multiplying these matrices together gives us a single matrix we can apply blindly without having to worry about knowing when to add a vector and when to multiply with a matrix

You can store ALL transformations in one matrix, rather than a matrix and a vector. Also, matrices are associative. This means a translation, then a rotation, is not the same as a rotation and then a translation. The translation could be affected by the previous rotations, or scales. This could be useful.

Also, when multiplying a 4*4 matrix with a 4 component vector, we could make the forth component of the vector 0, meaning it'll ignore the translation. A vector with a forth component of 0 means it's normalised. We don't want a normalised vector to me translated.

Yes, you could do all of this without matrices. It's just simpler and faster this way. You especially wouldn't want to manipulate a game with half matrices and half vectors.

• I'm know that rotation => translation is not the same as translation => rotation, but I can't relay see how this answers why it's done using multiplication and not addition. – GAX Jan 16 '14 at 9:59
• Because rotation and scale are not done through addition. We can't do a multiplication AND an addition in the same operation. – Ben Jan 16 '14 at 10:01
• I think you mean "matrix multiplication is non-commutative" (or, in general, affine transformation is non-commutative) rather than "matrices are associative". – Peter Taylor Jan 16 '14 at 16:29