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The following image taken from one of the books:

enter image description here

As it says, it moves to the dotted line which means box will move to the $x$ as well as $y$ direction.

In the same book, there's a vector product example as below, but I don't understand what $0$ means. I understand math, but not physics. Does 0 mean box won't move at all(but this contradicts the first picture). What does 0 mean?

enter image description here

Qmechanic
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Giorgi Lagidze
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    This is strange: normally we add the two vectors for forces to see what the resultant force will be, we do not take the dot product of the two. They are probably trying to tell you that the forces are along independent directions so that it's easier to predict what will happen. You can imagine if vector A was tilted downward then it would oppose the force from B a little bit, which would make things a little bit harder – Quantum Mechanic May 22 '23 at 13:09
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    why down vote this question? There are new students who don't understand the concepts, yet, and there is also an excess of poorly written books confusing said students. – JEB May 22 '23 at 14:22
  • This may help - What does the dot product of two vectors actually represent intuitively? What is its true meaning conceptually? – mmesser314 May 22 '23 at 14:42
  • In the first picture, you would add the vectors. The dot product would not be used here. The second picture looks a lot like the first picture, but it is used for something else. Perhaps they are talking about finding components of a vector. The components are useful for finding the direction of the total force, and for the direction of motion. – mmesser314 May 22 '23 at 14:45

2 Answers2

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What $\vec A \cdot \vec B = 0$ means is that the two forces are neither opposing nor assisting each other.

Note that it's a scalar value, with units of square-newtons, and is not something you'll see much in physics.

That it's a scalar means it has no information about direction of motion.

JEB
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  • So I should use dot product to also figure out the final force direction ? – Giorgi Lagidze May 22 '23 at 13:23
  • no, a scalar provides no directional information. – JEB May 22 '23 at 13:24
  • sorry I meant cross product – Giorgi Lagidze May 22 '23 at 13:26
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    Or maybe i should just add them – Giorgi Lagidze May 22 '23 at 13:27
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    @GiorgiLagidze always add two force vectors to get the combined force vector... – Prof. Legolasov May 22 '23 at 13:34
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    A cross product between a "pure" x vector (x 0 0) and a "pure" y vector (0 y 0) will result in a vector with only a z component, namely (0 0 xy). This vector has no physical meaning in this context, that is, it does not represent any force. As mentioned, just add the two force vectors together to obtain the total force acting on the body. – John Doe May 22 '23 at 13:35
  • @GiorgiLagidze well with a cross product you still have a unit problem (what's a square newton); moreover: a vector (e.g.: direction) can't be proportional to a cross product of two vectors, nor can an acceleration direction depend on the product of two forces, as it is not time reversiable. – JEB May 22 '23 at 13:40
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When the dot produt of two vectors is $0$, they are perpendicular to each other. The fomula in the picture is a special case of

\begin{equation} \vec{A} \cdot \vec{B} = |A||B| \cos \theta, \end{equation} where $\theta$ is the angle formed by the vectors. If this angle is $90^\circ$, like in the picture, the dot product is $0$, and the forces act on perpendicular directions. Therefore the box will move, and the total motion will be that caused by force $\vec{A}$ added to that caused by force $\vec{B}$. Since the forces have equal magnitudes, you get that the box will move along the dotted line.

A way to see this clearly is to add the forces before finding the motion of the box. Since $\vec{A} = (10 N) \vec{x}$ and $\vec{B} = (10 N) \vec{y}$, where $\vec{x}$ and $\vec{y}$ are unit vectors pointing on the $x$ and $y$ directions, respectively, the total force will be \begin{equation} \vec{F} = \vec{A} + \vec{B} = (10 N) \vec{x} + (10 N) \vec{y} = (10 N) (\vec{x} + \vec{y}). \end{equation} The vector $\vec{x} + \vec{y}$ points along the dotted line in the picture, the diagonal, so that is the direction along which $\vec{F}$ pushes the box.

If this is not clear, you can convince yourself by taking the dot product of $\vec{x} + \vec{y}$ it with $\vec{x}$ and $\vec{y}$ and use the formula in the top, to find exactly what angle it makes with these two vectors. You should find that in both cases the angle is $1/\sqrt{2}$ radians, or $45^\circ$.

Bairrao
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