Why Use Dot Product vs. Cross Product for Vector Calculations?

[Nikhil faraday]

Why some quantities are multiplied by using dot product and some by cross product ? Like work done is calculated by scalar product of force and displacement while torque is calculated by vector product of force and position vector?? Why do we give directions to quantities like angular velocity , angular acceleration? What is the significance of giving them directions? HELP !

 

[vanhees71]

I hope, it's intuitively clear to you that angular velocity and angular acceleration have to do with rotations. For anular velocity it's very simple to see since its magnitude is just the angle something is rotating per unit time. Now that doesn't tell you everything about the rotation but you need to know around which axis the thing is rotating, and that's given by a vector. So the angular velocity is vector $\vec{\omega}$, but it has not the meaning of vectors like a usual velocity of giving a direction something is "displaced" from one position to another but it indicates rotation. So you need a rule how to define the rotation axis in terms of a vector, and that's the right-hand rule. It says to get the rotation, point the thumb of your right hand in direction of $\vec{\omega}$. Then your fingers give the sense of rotation.

To dinstinguish these two kinds of vectors one usually calls the vectors describing displacements (or translation) as polar vectors and those indicating a rotation in the above sense axial vectors. Thus quantities related to relations like angular velocity, angular momentum, torque etc. are axial vectors.

As you also realized, the cross product has something to do with rotations too. That's a bit more involved. To explain it I'd need calculus, and I'm not sure whether you already have learned this.

 

[stevendaryl]

Angular velocity (or angular momentum) is not just a number, it has an associated "sense of rotation". If you set a bicycle wheel spinning, the spinning is characterized by: (1) the plane it is spinning in, (2) the rate that it is spinning (how many times per second does it spin completely around). For spinning in 3-dimensional space, you can use a vector, \vec{\omega} to convey these two pieces of information. The direction of \vec{\omega} is the axis of rotation (which is perpendicular to the plane of rotation), and the magnitude of \vec{\omega} is proportional to the rate of spinning.

By the way, angular velocity being represented as a vector with 3 components is kind of quirk of 3-dimensional space. In 2 dimensions, angular velocity has only one component. In 3 dimensions, it has 3 components. In 4 dimensions, it has 6 components. The general formula is that it has \frac{n(n-1)}{2} components in n-dimensional space.

 

[Nugatory]

Why some quantities are multiplied by using dot product and some by cross product ?

Asking why sometimes we use the dot product and sometimes we use the cross product is like asking why sometimes we use a hammer and sometimes we use a screwdriver - they're different tools used for different purposes, and you'll find both of them in a well-stocked toolbox.

There are lots of things you can do to combine two vectors; you can calculate the dot product, you can calculate the cross product, you can add them together, you can subtract one from the other. These are four different tools used for four different purposes, and with practice you develop a sense of which one works to describe a particular physical situation. Chances are that you've already learned to recognize problems that call for vector addition - vector addition is easy to visualize. The dot product and the cross product are a bit harder to visualize and come a bit later in your studies, but like a hammer and a screwdriver, the more you see them being used the better you'll get at recognizing situations that call for one or the other.

Why do we give directions to quantities like angular velocity , angular acceleration? What is the significance of giving them directions?

Let's use angular velocity as an example, because it's easy to visualize. I have a ball, and it's spinning around a vertical axis at 1000 RPM. I have another ball and its spinning about a horizontal axis at 1000 RPM. They have the same amount of spin, but it's in different directions; to completely describe the spin of the two balls I need both a number (1000 RPM) and direction (vertical versus horizontal). So your question "Why do we give directions to quantities like angular velocity" is backwards - we don't "give" them direction, they already have direction, so we are forced to use mathematical descriptions that include that direction.

(As an aside, this trick of using vectors to represent rotations and angular velocities and the like only works in three dimensions. However, we live in a three-dimensional world, so it works for us - and all that's we require of our tools).
[Edit: and thanks to @stevendaryl for reminding me of this]