Vector & basic operations

What is a vector

Definition: vector is a MAGNITUDE with a DIRECTION

Notation:

• v as a vector.
• |v| or ||v|| as its Magnitude, or Length, or Distance, or Absolute value, same idea
• Slope or angle as its Direction.
• (a, b) the two position there are called X-component & Y-component.

It’s not hard to understand the basic ideas of a vector, which consists of very basic knowledges form what we learned previously in high school:

• Magnitude of vector: is the same with calculating the distance of two points
• Direction of vector: is the same with calculating the slope of a line.

Distance vs. Displacement

• Distance is a scalar (“3 km”)
• Displacement is a vector (“3 km Southeast”)

Speed vs Velocity

• Speed is how fast something moves.
• Velocity is speed with a direction.

UNDERSTAND VECTOR'S MOVEMENT

Now we got a vector start from (-3,8) to the point (4, 5).
We say this vector has a magnitude ||v|| = √40 and has a direction to the bottom right.
But we could also represent the vector as only one point with two components: (7, -3).
Which has a HIDDEN INFORMATION that it start from the origin and direction is to the point (7, -3).
Are they different vectors? NO! They're the same vector, just starts at different places.

Because vector doesn’t specify where it starts. A vector only means A magnitude & a direction.

That’s why we could represent a vector in two ways:

• v is (-3,8) to (4, 5)
• v = (7, -3)

IT’S VERY IMPORTANT TO UNDERSTAND THIS IDEA, SO THAT WE COULD FURTHER UNDERSTAND WHY WE COULD MOVE VECTORS, AND BY SO WE COULD DO SUMS AND MULTIPLICATIONS AND SO ON

Adding & subtracting vectors

“Linear algebra is built on these operations v + w and cv: Adding vectors and multiplying by scalars." - Gilbert Strang

Adding & subtracting could be seen as a movement to a vector, or say how it travels.

Refer to Maths is fun.

• Adding vectors:

• Subtracting vectors: is just ADDING a NEGATIVE VECTOR

Understand vector's addition & subtraction

Refer to _Intro to Linear Algebra by Gilbert Strang: 1.1_.

“You can’t add apples and oranges.”
But you can add fruits!
Imagine you have one bag of fruits (3 apples, 4 oranges), and another bag of fruits (1 apple, 2 oranges).
So adding them together you will get one big bag of fruits (4 apples, 6 oranges),
from this big bag you could also split a smaller bag of fruits, and then you will call it subtraction.

A vector is the very idea of a bag of fruits.

Scalar Multiplication

It’s the same with the section DILATION or Scaling in Geometry.

Just for review of dilating a graph geometrically:
When you scale a graph by a factor:

• Every line of the graph scale by the SAME factor
• Every line will be PARALLEL to its origin line.

Refer to previous note: Transformation.

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