Concept: Multiplying a vector by a scalar quantity

Question

A scalar is a quantity which has the only magnitude $\|m\|$ in contrast to a vector which has both direction and magnitude $\|m\|\angle\theta$. Intuitively, multiplying a vector by a scalar scales a vector by the respective magnitude. Moreover, various texts claim, that multiplying a vector by scalar only changes the magnitude but direction remains unaffected.

An unexpected result is noticed when we multiply a negative scalar by a vector. Notably, a vector $\vec v = (x, y)$, when its multiplied by a negative scalar $-\|q\|$, it not only scales the magnitude by a quantity equals to $\|q\|$, but also flips the vector $-\|q\|\vec v = (-\|q\|x, -\|q\|y) = \|q(x^2+y^2)\|\angle{\left(\pi+\arctan\frac{y}{x}\right)}$.

Many notable texts, claim the behavior to be valid. But, I find a few contradictions

1. If by multiplying a vector $\vec v$ scales a vector, what is the intuition of negative scaling?
2. If the magnitude is absolute, and scalar has the only magnitude, how can scalar be negative?
3. If multiplying by a scalar does not change the direction, why is multiplying a vector by a negative quantity and thus flipping it, a valid behavior?

The way, I am trying to understand is, multiplying by a negative scalar is actually two operations.

1. Multiplying the vector by (-1) which flips a vector. This generates the negative inverse of a vector $\because \vec v + (-1)\times \vec v = 0$
2. Scale the resultant vector by the appropriate scalar quantity.

So, scalar -q is actually $\|q\| \times -1$ and thus multiplying with the vector $\vec v = \|v\|\angle\theta$

$$\Rightarrow -q \times \vec v = \|q\| \times -1 \vec v = \|q\| \times \|v\|\angle\left(\pi + \theta\right)$$

But, I cannot find a source to substantiate my reasoning and need the community support to help me with my understanding.

Answer 1

We have that for $\lambda\in \mathbb{R}$

• for $\lambda>0$ we have that the operation $\lambda\vec v$ scale $\vec v$ of a factor $|\lambda|$ with the same orientation

• for $\lambda<0$ we have that the operation $\lambda\vec v$ scale $\vec v$ of a factor $|\lambda|$ with the opposite orientation

Note that in both cases the direction doesn't change since both vectors belong to the same line.

Answer 2

In a real vector space where the scalars are real numbers, a scalar has a magnitude and a sign.

Thus when you multiply a vector by a scalar you scale the magnitude of the vector and change or not change the direction of your vector based on the sign of the scalar.

( The vector stays on the same line, which sometimes is interpreted as not changing the direction )

The concept is clear but the terminology is sometimes confusing.