Vectors multiplication
Vector multiplication is FUNDAMENTAL skill to solve
Matrices multiplication.
THE VERY FIRST THING TO DO WITH A VECTOR MULTIPLICATION OR MATRIX MULTIPLICATION, IS TO FORGET EVERYTHING ABOUT ARITHMETIC MULTIPLICATION !!
OTHERWISE, YOU WILL FALL INTO AN ENDLESS CONFUSION!
Just to know, multiplication of vectors or matrices, AREN’T really multiplication, but just look like that. You can see them as operations to get SOMETHING.
There’re two operations are called multiplication for vectors:
Dot product: express asV₁ · V₂, named for the dot symbol. It's meant to get theProduct of two magnitudes.Cross product: express asV₁ × V₂, named for the cross symbol. It's meant to get anew vector.
The
cross productis very very very limited in use, and NOT as fairly often in use as thedot product. So don't waste time on this unless having certain use of it.
Boosting
IT’S THE VERY CORE SENSE OF MAKING A MULTIPLICATION OF VECTORS OR MATRICES.
Multiplication ISN’T just Repeat counting in Arithmetic anymore.
Not 4×3 = 4+4+4 anymore!
It’s rather kind of Growth, or empowerment, or boosting.
We'd say we tripled 4, or say number 4 grow with speed of 3, or to say number 4 grows with a boosting of 3.
Whatever you'd say, you get the idea.
Multiplication a process of double, triple, quadruple ....
JUST TO REMEMBER: FORGET ABOUT ARITHMETIC MULTIPLICATION, ALWAYS SEE MULTIPLICATION AS BOOSTING.
Dot product
REMEMBER: A DOT PRODUCT DOESN’T GIVE YOU A VECTOR, BUT ONLY A NUMBER, A SCALAR, A PRODUCT OF TWO MAGNITUDES.
The purpose:
It is NOT to get a new vector, and NOT toReduce dimension,
its only purpose IS to get a quantity, a magnitude, a number!
For an intuitive video refer to Khan academy physics: Dot Product.
For more explains in detail: Vector Calculus: Understanding the Dot Product
Maths is fun: dot product.
3Blue1Brown: Dot products and duality | Essence of linear algebra
Understand Dot product in business
Refer to _Intro to linear algebra by Gilbert Strang: 1.2_.
Understand Dot product in physics
It makes lots more sense to think
dot productin physics way than maths algebraic way.
Just to think Two forces "a & b" are pulling a box,
so how much power did it pulled on the direction of a, or how much on the direction of b?
Vectors on same direction
Let’s make it easier before digging in:
assume there’s no angle, Two forces "a & b" are pulling to the same way, the same direction,
so how much power would it be pulled?
Well, the force a & b working together, it's a process of Boosting the energy!
It's not ADDING together anymore, it's BOOSTING!
Let's say the force a has 3 units power, b has 6 units power.
So every 1 unit power a pulls, b will pull 2 units power.
Then it make sense:
The total power pulling the thing would be 3 · 6 = 18 units
Vectors on different direction
So the Two forces AREN'T pulling the box at the same direction anymore, how much power did it pulled on the direction of a, or how much on the direction of b?
Let’s think about how much power it’s pulling on the direction of b.
Since a is pulling on a bit wrong way, so a's power ISN'T 100% working on b's way.
How much power left there?
It depends on the angle.
So to calculate how much left, we use |a| × cos(θ),
and we got a PROJECTION or a reflection or a shadow of a on b!
Then it become like this picture again:
How amazing it is!
And now we could Boost the power on b: |b| × |a|×cosθ
Ways of calculating dot product
There’re two ways to calculate the dot product (I made up the names):
- Shadow Boost:
- Axes Boost:
Result of two ways are SAME.
Remember: Boosting is not working when two vectors are Perpendicular, which product is
0.
Shadow Boost
We reflect one vector on another one, then Boost the energy.
Intuition:
Axes Boost
We break two vectors to
X-axisandY-axis, and BOOST on each axis.
Easier to remember the formula is:
Intuition:
Examples:
Example
Dot product & Symmetry
Dot product has a relationship with Symmetry.
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