Root of any degree Root(n, x)

Notation:
Root(n, x), where
n – redicand
x – index
x, n – any numbers or expressions.
For an even root, the redicand cannot be negative.

Examples:
Cubic root of fraction 2/5
Root(2/5, 3)

Other examples
Root(1.5, 3)
Root((3*5), 3/2)
Root(1.5, 3/7)

Exponentiation Pow(n, x)

Notation:
Pow(n, x), where
n – base
x – exponent or power
x, n – any numbers or expressions.

Examples:
Five to the power of three
Pow(5, 3)

Other examples
Pow(12.5, 3)
Pow((3-5), 3/2)
Pow(1.5, Sqrt(2))

Logarithm Log(n, x)

Notation:
Log(n, x), where
n – the number whose logarithm you want to find
x – base.
x > 0, x ≠ 1, n > 0

Examples:
Log5 34 (logarithm of 34 to base 5), should be written as
Log(34, 5)

Natural logarithm Ln(n)
The base is equal to Euler's number e
(e = 2.7182818284...)

Notation:
Ln(n), where
n > 0<

Examples:
Ln(7)

Common logarithm Lg(n)
The logarithm base 10

Notation:
Lg(n), where
n > 0

Examples:
Lg(1.6)

Binary logarithm Lb(n)
The logarithm base 2

Notation:
Lb(n), where
n > 0

Examples:
Lb(3/6)

Greatest common divisor Gcd(n, m)

Notation:
Gcd(n, m), where
n, m – non-negative integers

Examples:
Gcd(12; 16) should be written as
Gcd(12, 16)

Least common multiple Lcm(n, m)

Notation:
Lcm(n, m), where
n, m – non-negative integers

Examples:
Lcm(4; 23) should be written as
Lcm (4, 23)

Trigonometric functions
All trigonometric functions take one or two arguments. If the function takes one argument, then the number is considered to be in radians.

Sine of an angle Sin(x)

Notation:
Sin(x)
Sin(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Sine π/3 radians
Sin(π/3) or Sin(π/3, Rad)

Sine 60° degrees
Sin(60, Deg)

Cosine of an angle Cos(x)

Notation:
Cos(x)
Cos(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Cosine π/3 radians
Cos(π/3) or Cos(π/3, Rad)

Cosine 60° degrees
Cos(60, Deg)

Tangent of an angle Tan(x)

Notation:
Tan(x)
Tan(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Tangent π/3 radians
Tan(π/3) or Tan(π/3, Rad)

Tangent 60° degrees
Tan(60, Deg)

Cotangent of an angle Cot(x)

Notation:
Cot(x)
Cot(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Cotangent π/3 radians
Cot(π/3) or Cot(π/3, Rad)

Cotangent 60° degrees
Cot(60, Deg)

Secant of an angle Sec(x)

Notation:
Sec(x)
Sec(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Secant π/3 radians
Sec(π/3) or Sec(π/3, Rad)

Secant 60° degrees
Sec(60, Deg)

Cosecant of an angle Csc(x)

Notation:
Csc(x)
Csc(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Cosecant π/3 radians
Csc(π/3) or Csc(π/3, Rad)

Cosecant 60° degrees
Csc(60, Deg)

Inverse trigonometric functions
All inverse trigonometric functions take one or two arguments. If the function takes one argument, then the function will return the solution in radians.

Arcsine Asin(x)

Notation:
Asin(x)
Asin(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Arcsine 1/3 (get the solution in radians)
Asin(1/3) or Asin(1/3, Rad)

Arcsine 1/3 (get the solution in degrees)
Asin(1/3, Deg)

Arccosine Acos(x)

Notation:
Acos(x)
Acos(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Arccosine 1/3 (get the solution in radians)
Acos(1/3) or Acos(1/3, Rad)

Arccosine 1/3 (get the solution in degrees)
Acos(1/3, Deg)

Arctangent Atan(x)

Notation:
Atan(x)
Atan(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Arctangent 1/3 (get the solution in radians)
Atan(1/3) or Atan(1/3, Rad)

Arctangent 1/3 (get the solution in degrees)
Atan(1/3, Deg)

Arccotangent Acot(x)

Notation:
Acot(x)
Acot(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Arccotangent 1/3 (get the solution in radians)
Acot(1/3) or Acot(1/3, Rad)

Arccotangent 1/3 (get the solution in degrees)
Acot(1/3, Deg)

Arcsecant Asec(x)

Notation:
Asec(x)
Asec(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Arcsecant 1/3 (get the solution in radians)
Asec(1/3) or Asec(1/3, Rad)

Arcsecant 1/3 (get the solution in degrees)
Asec(1/3, Deg)

Arcsecant Acsc(x)

Notation:
Acsc(x)
Acsc(x, measure)
Where
x – number
measure – can take values Rad or Deg

Examples:
Arcsecant 1/3 (get the solution in radians)
Acsc(1/3) or Acsc(1/3, Rad)

Arcsecant 1/3 (get the solution in degrees)
Acsc(1/3, Deg)

Expressions containing multiple embedding of functions and mathematical operations
Any expression can contain other expressions, the limit on the length of an expression is 100 characters.

Examples:
Root(Pow(3, 6), 2);
(5/2-4)*34/5-(Root(3, 2))
(12-123+5)/(12.45*(34/6))
Sin(60, Deg)+Cos(45, Deg)
etc.

Specify the form of the first vector
Specify the form of the second vector
Specify the form of the third vector

Enter the coordinates of the first vector
a̅ = { ; ; }

Enter the coordinates of the second vector
b̅ = { ; ; }

Enter the coordinates of the third vector
c̅ = { ; ; }

How to Find Mixed Product of Vectors

Example #1
Let's find the mixed product of vectors in three-dimensional space. Vector coordinates are given by points.
Coordinates of point A of vector AB: (7; 9 ; -7)
Coordinates of point B of vector AB: (-1 ; 0 ; 16)
Coordinates of point C of vector CD: (-4 ; -6 ; 2)
Coordinates of point D of vector CD: (3 ; 0 ; 9)
Coordinates of point E of vector CD: (14 ; -1 ; 2)
Coordinates of point F of vector CD: (6 ; -3 ; -4)

Step by step solution:

Let's calculate the coordinates of the first vector from two points A and B:

AB = {x_B - x_A; y_B - y_A; z_B - z_A} = {-1 - 7 ; 0 - 9 ; 16 - (-7)} = {-8 ; -9 ; 23}

Let's calculate the coordinates of the second vector from two points C and D:

CD = {x_D - x_C; y_D - y_C; z_D - z_C} = {3 - (-4) ; 0 - (-6) ; 9 - 2} = {7 ; 6 ; 7}

Let's calculate the coordinates of the third vector from two points E and F:

EF = {x_F - x_E; y_F - y_E; z_F - z_E} = {6 - 14 ; -3 - (-1) ; -4 - 2} = {-8 ; -2 ; -6}

The mixed product of vectors is a number equal to the value of the determinant of the third-order matrix, the rows of which are the coordinates of the vectors being multiplied. Let's find the value of the determinant:

AB ⋅ CD ⋅ EF = ^{^{AB_x AB_y AB_z
CD_x CD_y CD_z
EF_x EF_y EF_z}} = AB_x^{CD_y CD_z
EF_y EF_z} - AB_y^{CD_x CD_z
EF_x EF_z} + AB_z^{CD_x CD_y
EF_x EF_y} = AB_xCD_yEF_z - AB_xCD_zEF_y - AB_yCD_xEF_z + AB_yCD_zEF_x + AB_zCD_xEF_y - AB_zCD_yEF_x

AB_x = -8
AB_y = -9
AB_z = 23
CD_x = 7
CD_y = 6
CD_z = 7
EF_x = -8
EF_y = -2
EF_z = -6

AB ⋅ CD ⋅ EF = ^{^{(-8) (-9) 23
7 6 7
(-8) (-2) (-6)}} = (-8)^{6 7
(-2) (-6)} - (-9)^{7 7
(-8) (-6)} + 23^{7 6
(-8) (-2)} = (-8 ⋅ 6 ⋅ (-6)) - (-8 ⋅ 7 ⋅ (-2)) - (-9 ⋅ 7 ⋅ (-6)) + (-9 ⋅ 7 ⋅ (-8)) + (23 ⋅ 7 ⋅ (-2)) - (23 ⋅ 6 ⋅ (-8)) = 288 - 112 - 378 + 504 + (-322) - (-1104) = 1084

Example #2
Let's find the mixed product of vectors in three-dimensional space.
Vector coordinates a: (5 ; 1 ; 7)
Vector coordinates b: (2 ; 4 ; 6)
Vector coordinates c: (3 ; 8 ; 9)

Step by step solution:

a ⋅ b ⋅ c = ^{^{a_x a_y a_z
b_x b_y b_z
c_x c_y c_z}} = a_x^{b_y b_z
c_y c_z} - a_y^{b_x b_z
c_x c_z} + a_z^{b_x b_y
c_x c_y} = a_xb_yc_z - a_xb_zc_y - a_yb_xc_z + a_yb_zc_x + a_zb_xc_y - a_zb_yc_x

a_x = 5
a_y = 1
a_z = 7
b_x = 2
b_y = 4
b_z = 6
c_x = 3
c_y = 8
c_z = 9

a ⋅ b ⋅ c = ^{^{5 1 7
2 4 6
3 8 9}} = 5^{4 6
8 9} - 1^{2 6
3 9} + 7^{2 4
3 8} = (5 ⋅ 4 ⋅ 9) - (5 ⋅ 6 ⋅ 8) - (1 ⋅ 2 ⋅ 9) + (1 ⋅ 6 ⋅ 3) + (7 ⋅ 2 ⋅ 8) - (7 ⋅ 4 ⋅ 3) = 180 - 240 - 18 + 18 + 112 - 84 = -32

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