# Vectors

### Coordinates of the vector A⃗B

#### Calculate the coordinates of the vector A⃗B pointing from the point A(x1,y1) to the point B(x2,y2)

Enter the coordinates of the point A(x1,y1)

x1: y1:

A (,)

Enter the coordinates of the point B(x2,y2)

x2: y2:

B (,)

#### Calculate the coordinates of the vector A⃗B pointing from the point A(x1,y1,z1) to the point B(x2,y2,z2)

Enter the coordinates of the point A(x1,y1,z1)

x1:
y1:
z1:

A (,,)

Enter the coordinates of the point B(x2,y2,z2)

x2:
y2:
z2:

B (,,)

### Dot product a⃗ ∙ b⃗

##### Find the dot product for the given vectors a⃗ = ( x1, y1,z1 ) and b⃗ = ( x2, y2,z2 )

Enter the coordinates of the vector a⃗ =(x1,y1,z1)

x1:
y1:
z1:

a⃗ = (,,)

Enter the coordinates of the vector b⃗ =(x2,y2,z2)

x2:
y2:
z2:

b⃗ = (,,)

#### a⃗ ∙ b⃗ = x1∙x2 + y1∙y2 + z1∙z2 = {{x41}} ∙ {{x42f()}} + {{y41}} ∙ {{y42f()}} + {{z41}} ∙ {{z42f()}} = {{(x41*x42+y41*y42+z41*z42)}}

##### The dot product of two Euclidean vectors a⃗ and b⃗ given the magnitudes and the angle between the vectors

Enter the magnitude of the vector ∣

∣ a⃗ ∣: {{x51aps()}}
∣ b⃗ ∣: {{y51aps()}}
the angle ∡ (a⃗,b⃗) = α
∡α = °
cosα={{cos4() }}

#### a⃗ ∙ b⃗ =∣ a⃗ ∣ ∣ b⃗ ∣ cos α = a⃗ ∙ b⃗ = {{x51}} ∙ {{y51}} ∙ {{cosf() }} = {{ob4 () }}

If you want to round out the result, enter the number of digits
a⃗ ∙ b⃗ = {{okrugao()}}

### Ugao između vektora

##### The angle between the vectors a⃗ and b⃗ , given the coordinates of the vectors and the dot product ∡ (a⃗,b⃗)

Enter the magnitude of the vector ∣

∣ a⃗ ∣: {{x61aps()}}
∣ b⃗ ∣: {{y61aps()}}
the dot product
a⃗ ∙ b⃗ =

#### ∡ (a⃗,b⃗) = arccos(a⃗ ∙ b⃗ /(∣ a⃗ ∣ ∣ b⃗ ∣)) = ∡ (a⃗,b⃗) = arccos( {{z61}} / ( {{x61}} ∙ {{y61}} ) ) = {{ okrugao2() }} °

If you want to round out the result, enter the number of digits
a⃗ ∙ b⃗ = {{okrugao1()}}

### The cross product of the vectors a⃗ x b⃗

##### Cross product of the vectors a⃗ and b⃗ given their Cartesian coordinates

Enter the coordinates of the vector a⃗ =(x1,y1,z1)

x1:
y1:
z1:

a⃗ = (,,)

Enter the coordinates of the vector b⃗ =(x2,y2,z2)

x2:
y2:
z2:

b⃗ = (,,)

```	|  i⃗  j⃗  k⃗  |
a⃗ x b⃗ =	| x1 y1 z1 |
| x2 y2 z2 |```

= i⃗ (y1z2 - y2z1) + j⃗ ( z1x2 - z2x1) + k⃗ ( x1y2 - x2y1) = {{y71*z72 - y72*z71}}i⃗ + {{z71*x72 - z72*x71}}j⃗ + {{x71*y72 - x72*y71}} k⃗

### Mixed product of the vectors a⃗∙( b⃗ x c⃗)

##### Mixed product of the vectors a⃗ , b⃗ and c⃗ given their Cartesian coordinates

Enter the coordinates of the vector a⃗ =(x1,y1,z1)

x1:
y1:
z1:

a⃗ = (,,)

Enter the coordinates of the vector b⃗ =(x2,y2,z2)

x2:
y2:
z2:

b⃗ = (,,)

Enter the coordinates of the vector c⃗ =(x3,y3,z3)

x3:
y3:
z3:

c⃗ = (,,)

```		| x1 y1 z1 |
a⃗( b⃗ x c⃗) =      | x2 y2 z2 |=
| x3 y3 z3 |```