Math cabinet Coordinates of the vector Dot product Cross product Mixed product SR
Coordinates of the vector Dot product Cross product Mixed product SR

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 (,)

A⃗B = u⃗ = ( x2 - x1 , y2 - y1 )
A⃗B = u⃗ = ( {{ x2}} - {{ x1f()}} , {{ y2}} - {{ y1f()}} ) = ( {{ x2 - x1 }}, {{ y2 - y1 }} )
A⃗B = u⃗ = {{ x2 - x1 }} i⃗+ {{ k2y2y1f() }} j⃗

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 (,,)

A⃗B = u⃗= ( x2 - x1 , y2 - y1 , z2 - z1 )
A⃗B = u⃗ = ( {{x32}} - {{x31f()}} , {{y32}} - {{y31f()}} , {{z32}} - {{z31f()}} ) = ({{ x32 - x31 }},{{ y32 - y31 }},{{ z32 - z31 }})
A⃗B = u⃗ = {{ x32 - x31 }} i⃗ + {{ ky2y1f()}} j⃗ + {{ kz2z1f() }} k⃗

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

a⃗ x b⃗ = ({{y71*z72 - y72*z71}},{{z71*x72 - z72*x71}},{{x71*y72 - x72*y71}})

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 |


a⃗∙( b⃗ x c⃗) = x1y2z3 + y1z2x3 + z1x2y3 - x3y2z1 - y3z2x1 - z3x2y1
a⃗∙( b⃗ x c⃗) = {{x81}} ∙ {{y82f()}} ∙ {{z83f() }}+ {{y81f()}} ∙ {{z82f()}} ∙ {{x83f()}} + {{z81f()}} ∙ {{x82f()}} ∙ {{y83f()}} - {{x83f()}} ∙ {{y82f()}} ∙ {{z81f()}} - {{y83f()}} ∙ {{z82f()}} ∙ {{x81f()}} - {{z83f()}} ∙ {{x82f()}} ∙ {{y81f()}} = {{x81*y82*z83 + y81*z82*x83 + z81*x82*y83 - x83*y82*z81 - y83*z82*x81 - z83*x82*y81}}