Vectors

Question 1

Vectors dot product

Answer 1

Dot product us a result of multiplication of 2 vectors
~~~
a = [1,2,3] b = [3,2,1]

a.b = 13 + 22 + 3*1 = 10
~~~

Question 2

Dot product with vector itself

Answer 2

Dot product with vector itself
a=[1,2,3]
a.a = 11+22+3*3= 10

why is this useful? because we can find magnitude(length) of vector

Length of vector is a square root of its dot product: |a| = sqr(10)

Question 3

Vector addition

Answer 3

1. Addition (substarction)
   To add the vectors (x₁,y₁) and (x₂,y₂), we add the corresponding components from each vector: (x₁+x₂,y₁+y₂)
   on coordinate system addition of 2 vectors is the same as connecting 2 vectors

Question 4

Vector multiplication by scalar

Answer 4

Vector scalar multiplication looks like stretching out the vector or shrinking it iw we multiply by scalar < 1
a = [2,1] multiply by scalar 3 => a = [23 ,13] = [6, 3]
we stretched out vector 4 times

Question 5

Unit vector

Answer 5

A vector is a quantity that has both magnitude, as well as direction. A vector that has a magnitude of 1 is a unit vector. It is also known as Direction Vector

If we get rid of length(magnitutde) vector is left with direction only.
how to find unit vector
simply divide vector by its magnitude.

a = [3,4] => |a| = sqr(9, 16 ) = 5 - this is magnitude of vector a

a> (>this shold be as a hat for a) = [3,4] /5 => [3/5, 4,5] this will be a unit vector for a

Question 6

Orthogonal vector

Answer 6

We say that 2 vectors are orthogonal if they are perpendicular (angle 90 degree)
to each other. i.e.** the dot product of the two vectors is zero**

 a = [4, 2, -1] b = [1, -3, - 2] 

a.b = 4*2 + 2*(-3) + (-3, -2) = 8 + 6 - 2 = 0 

a and be are orthogonal

Question 8

Orthonormal vectors

Answer 8

Rtonormal vector is a orthogonal vector for which the len is 1
Orthonormal: 1. Dot product is 0
2. Lenght/magnitude is 1

Question 9

Orthogonal matrix

Answer 9

Square matrix are considered orthogonal if their columns make up an orthonormal set of vectors
For orthogonal matrix O its inverse O -1 is the same as Ot (transpose) it makes very easy finding inverse of matrix. Instead of OO-1 = I we can use OOt=I