As we know, vectors have both direction and magnitude, and Accelerator's ability is Vector Change. I always believed that he could control both attributes of vectors, until my friends claimed that he could only change vectors' directions.
So is there any clear cut official statement to this question?
Some proof that he can control the magnitude of vectors:
I'm sure that there could be more examples which I can't remember off the top of my head.
I take the position that Accelerator cannot control the magnitude of vectors.
The Japanese for Accelerator's "vector manipulation" ability is: 向き【ベクトル】変換. The two fundamental properties of a vector are magnitude (大きさ) and direction (向き). If we ignore the furigana on 向き, Accelerator's ability translates as "direction changing" or "direction transformation". (Also, 向き isn't a synonym for vector in any standard usage, so there's clearly an emphasis on the "direction" aspect here.)
Consider the following quote from volume 15, with all furigana preserved:
一方通行【アクセラレータ】は何らかのベクトルを拾い、その向きを変換し、一点に集中して垣根帝督【ていとく】を攻撃【こうげき】している。
Roughly translated, this comes out to:
Accelerator brought together some manner of vectors (ベクトル), altered their direction (向き), brought them to a point, and attacked Kakine Teitoku.
We see here that there is an explicit reference to modifying the direction of vectors, and no reference to modifying the magnitude of vectors. This doesn't prove anything directly, but its absence is reasonably informative.
A cursory search across all 22 volumes of the initial light novel series (i.e. before New Testament) plus the two associated side story volumes reveals no cases in which 大きさ was mentioned within a few lines of ベクトル. (I might've missed something, though; this isn't all that great evidence. I'm just pointing out that there seem to be no references to the matter of magnitude either way.)
Direct physical evidence that Accelerator cannot control vector magnitudes cannot exist (you can't prove a negative). It is easy, however, to demonstrate a reasonably plausible physical mechanism by means of which Accelerator could do everything he does without needing to control the magnitude of vectors (as I will do below). By the principle of parsimony, it follows that we need not assume that Accelerator is able to control vector magnitudes.
We begin by looking at energetic considerations. Consider a 1 kg lump of iron that is at room temperature. Roughly speaking, the total thermal energy of that lump is on the order of 1e5 ~ 1e6 joules. This is sufficient energy to propel a 50 kg human at a velocity of ~100 mi/hr, or a 50 g stone at ~3000 mi/hr (~Mach 4). If Accelerator is in contact with the earth, he has access to an energy reservoir roughly 1e24 times larger. Conclusion: Accelerator has access to enough energy to do pretty much anything, if he has a way to harness some or all of the thermal energy of objects he is in contact with.
The question remains: can Accelerator harness thermal energy? I claim the answer is yes.
Let us assume for the sake of argument that the only thing Accelerator can do is take a moving particle and alter its direction of motion (we're going to handwave our way past things like wavefunctions, Heisenberg uncertainty, and [in particular] conservation of momentum, because obviously Accelerator couldn't exist in the real world, so some physical laws have to give). Using this framework, let's see how Accelerator can do some things.
In general, I don't think it's possible to formulate a valid argument of the form "Accelerator is capable of doing X, therefore he must be able to manipulate vector magnitudes". As long as Accelerator has access to warm matter (which he does unless you stick him in a vacuum), he has access to particles in motion, with their associated momentum vectors, which can be manipulated to do just about anything.
Take all of this with a grain of salt - as the perennial "can Flash beat Accelerator" debates (and their ilk) demonstrate, Accelerator's abilities are not well-defined in a physical sense. Vectors don't physically exist - they're merely a useful mathematical abstraction we use to describe certain features of the universe. Trying too hard to justify Accelerator's abilities in terms of physical principles is a recipe for failure. In the end, Accelerator's vector manipulation does whatever Kamachi wants it to do.
Accelerator's ability can be broken down into two parts: Redirection and Vector control.
Redirection is his defensive, passive posture which reflects any vector approaching his body.
Vector control is his active ability where he can choose to manipulate specific vectors by both magnitude and direction. This ability requires Accelerator to constantly be computing the combination of vectors he plans to use.
In the sisters arc, Accelerator demonstrated a few different types of vector control.
Vector shooting was when he kicked the rock at Touma. He was able to increase the speed of the rock and make it substantially faster than normal.
Near the end of the fight, he was also able to create this huge plasma field. This required him to gather wind in a specific location, and then further compressing the wind (wind control and plasma storm).
There was also Earth Stomp where Accelerator simply just stomps on the ground and controls the vectors to create a small rock attack upon the enemy. Along with that, Accelerator uses the Earth to propel himself forward in other situations.
As the story progresses, Accelerator has a new arsenal of moves that require the manipulation of magnitude, however those were not covered in the Anime yet. As of the sisters arc in both Index and Railgun, these are the different types of vector control that have been displayed.
