About vectors, how does Dot Product help with anything? My teacher says it's amazingly useful, but I'm like ?!

Answer 1

For example it makes it easy to compute the angle between two vectors. If the angle is theta then cos(theta)=a . b/(|a||b|) where |a| and |b| denote the length of the vectors a and b respectively. Or it lets you compute lengths of vectors:
|a|=sqrt{a .a}
for example. So many things you may want to know about vectors, or analytic geometry in general, can be expressed in terms of dot product. Now is it the most useful thing abot vectors? I doubt it. Should you care about them? Well if you are going to work with vectors at any level, then yes.

Answer 2

Don't have enough information to answer your question, but there is a field of Mathematics called "Vector Algebra" which is based upon Trigonometry.

You can add and subtract Vectors in Rectangular form and Multiply vectors in Polar Form.

For example let us take the 3,4,5 triangle and express the Hypotenuse in Rectangular form.

The Vertical excursion is the Opposite side and the Horizontal Excursion is the Adjacent Side of the Right Triangle.

A sqared Plus B squared = C squared.

or "C" (The Hypotenuse) = The square root of
A squared Plus B squared

That is an example of Rectangular notation.

the Angle of that Right Triangle is 53.1 Degrees when A = 3 and B = 4.

in Polar notation: 5 at an angle of 51.3 deg.

This type of Algebra can be used to find the angle of a flight of stairs, the Height of a Building and also to solve seris and parallel Electronic Circuits.

When solving problems using these methods, it is necessary to add and subtract Vectors as well as Multiply them. All the rules are too lenghy to enter here.

Answer 3

In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space

Also in math it is very useful in determining if two vectors are orthogonal to one another.

Answer 4

dot product is also known as the scalar product and inner product

in not so laymans terms if you want the geometric interpretation of vector X dot Vector Y it would be the length of the projection of vector X onto unit vec Y~ when the 2 vectors are placed so that their tails coincide

the dot product of 2 perpendicular vectors is 0

the dot product of a vector with itself should give you the norm or 'length' of the vector [there's one of the most important aspects of the dot product]

Answer 5

In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space.

The dot product of two vectors (from an orthonormal vector space) a = [a1, a2, … , an] and b = [b1, b2, … , bn] is by definition:

n
Σ ai bi = a1 b1 + a2 b2 + a3 b3 + .................an bn
i = 1

Answer 6

Dot product lets you multiply two vectors very easily.

If you have two vectors, let's call them (a,b,c) and (r,s,t). To multiply the two vectors, you just take the "dot product" which is simply multiplying like thus: (a*r,b*s,c*t)

Is that the kind of info you were looking for?

Answer 7

You can calculate the work done by a force. Its the force dotted with the displacement vector.

Answer 8

The mechanics of calculating dot products are straightforward. There are applications in every branch of science and engineering. Some of my favorite applications are
1. calculating molecular bond lengths and angles and interactions between molecules (eg dipole-dipole interactions are important in all biological systems.)
2. solving crystal structures
3. design of optical devices
4. pattern recognition. Objects can be compared by computing the dot product of multidimensional data vectors.
Ask any scientist or engineer and you will get a different set of applications. I would advise you to learn linear algebra too.

Answer 9

I use dot product often in my engineering work. For example, Gauss' law of electrostatics uses dot product to express a physical relationship.

Think of dot product like a tool. You get it out, learn how it works, put it back in your toolbox, and then when the time comes when you need that one special tool it's there for you to use.

Answer 10

luk here ... spose u got 2 vectors like force n displacement. u need 2 find da work done by the force on the object 2 find the displacement produced in it. 4 dat u need da dot product ... yup its dat important. it may seem just a dot ... its hell lotta more important.