Notice

We and selected partners use cookies or similar technologies as specified in the cookie policy.
You can consent to the use of such technologies by closing this notice, by scrolling this page, by interacting with any link or button outside of this notice or by continuing to browse otherwise.
To find out more about the categories of personal information collected and the purposes for which such information will be used, please refer to our privacy policy.

California Consumer Notice

 0 AC +/- รท 7 8 9 ร 4 5 6 - 1 2 3 + 0 00 , =

# Vector Triple Product Calculator

## This calculator calculates the triple (mixed) product of vectors using the given coordinates and gives a detailed solution to all stages of the calculation.

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 = {xB - xA  ; yB - yA; zB - zA} = {-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 = {xD - xC  ; yD - yC; zD - zC} = {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 = {xF - xE  ; yF - yE; zF - zE} = {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 โ CD โ EF =
 ABx ABy ABz CDx CDy CDz EFx EFy EFz
= ABx
 CDy CDz EFy EFz
- ABy
 CDx CDz EFx EFz
+ ABz
 CDx CDy EFx EFy
= ABxCDyEFz - ABxCDzEFy - AByCDxEFz + AByCDzEFx + ABzCDxEFy - ABzCDyEFx
ABx = -8
ABy = -9
ABz = 23
CDx = 7
CDy = 6
CDz = 7
EFx = -8
EFy = -2
EFz = -6
AB โ CD โ 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 โ 6 โ (-6)) - (-8 โ 7 โ (-2)) - (-9 โ 7 โ (-6)) + (-9 โ 7 โ (-8)) + (23 โ 7 โ (-2)) - (23 โ 6 โ (-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:

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:
a โ b โ c =
 ax ay az bx by bz cx cy cz
= ax
 by bz cy cz
- ay
 bx bz cx cz
+ az
 bx by cx cy
= axbycz - axbzcy - aybxcz + aybzcx + azbxcy - azbycx
ax = 5
ay = 1
az = 7
bx = 2
by = 4
bz = 6
cx = 3
cy = 8
cz = 9
a โ b โ 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 โ 4 โ 9) - (5 โ 6 โ 8) - (1 โ 2 โ 9) + (1 โ 6 โ 3) + (7 โ 2 โ 8) - (7 โ 4 โ 3) = 180 - 240 - 18 + 18 + 112 - 84 = -32
 You may also find the following calculators helpful Linear Algebra and Analytical Geometers Calculators Matrix Addition and Subtraction Calculator Matrix Multiplication Calculator Matrix Transpose Calculator Matrix Determinant Calculator Inverse Matrix Calculator Length of a Line Segment. Distance Between Points Online Calculator. Vector Coordinates Calculator From Two Points Vector Modulus (Length) Calculator Vector Addition and Subtraction Calculator Vector Dot Product Calculator. By Using the Length of the Vectors and Cosine of the Angle Between Vectors. Vector Dot Product Calculator. By Using Vector Coordinates. Vector Product Calculator. By Using Vector Coordinates. Vector Triple Product Calculator Vector Multiplication by Number Calculator Angle Between Vectors Calculator Vector Collinearity Check Calculator Vector Coplanarity Calculator Combinatorics Calculators Permutations of N Elements Calculator Combinations of N Elements by M Calculator Permutations of N Elements by M Calculator Numeral Systems Calculators Roman Numeral Calculator Roman Numerals From 1 to 3999 Numeral Systems Calculator Binary Calculator. Addition, Subtraction, Multiplication and Division of Binary Numbers. Fraction Calculators Simplifying Fractions Calculator Improper to Mixed Fraction Calculator Mixed Fraction to Improper Calculator Fractions Calculator With Step by Step Solution Raising a Fraction to a Power. Online Calculator. Decimal to Fraction Calculator Fraction to Decimal Calculator Compare Fractions Calculator Common Denominator Calculator Trigonometers Calculators Sine Calculator Cosine Calculator Tangent Calculator Cotangent Calculator Secant Calculator Cosecant Calculator Arcsine Calculator Arccosine Calculator Arctangent Calculator Arccotangent Calculator Arcsecant Calculator Arccosecant Calculator Calculators (Number theory) Mathematical Expressions Calculator Calculator Parentheses Prime Factorization Calculator Gcd and Lcm Calculator With Steps Gcd and Lcm calculator using Euclidean algorithm Gcd and Lcm Calculator for Any Number of Numbers Break Down the Number Into Its Place Value Divide a Number by a Ratio. Online Calculator With Steps Percentage Calculator With Steps Scientific Notation to Decimal Conversion Calculator Scientific Notation Converter Factorial Calculator With Steps Logarithm Calculator Quadratic Equations Calculator With Steps Remainder Calculator Root Calculator Decimal Period Calculator Big Number Calculator Rounding Numbers Calculator Properties of Roots and Exponents Calculator Complex Number Calculator Area Calculators Area of a Square Calculator Area of a Rectangle Calculator Math Trainers Addition Math Trainer Subtraction Math Trainer Multiplication Math Trainer Division Math Trainer Multiplication Table Math Trainer Online Preschool Counting Trainer Online Preschool Mindfulness Trainer Online Addition Subtraction Multiplication Division Math Trainer. Find the Correct Answer. Online Numeric Expression Trainer Converters Length Converter Speed Converter Acceleration Converter Physics Calculators Mechanics Speed Time Distance Calculator Acceleration Speed Distance Calculator Displacement Time Calculator Time Calculator Newton's Second Law Calculator Gravitational Force Calculator Momentum Calculator Impulse Calculator Object Weight Calculator Optics Light Reflection and Refraction Calculator Electricity and Magnetism Ohm's Law Calculator Coulomb's Law Calculator Electric Field Strength Calculator Point Electric Charge Calculator Q Force Acting on Charge Calculator Distance From Charge Calculator Potential Charge Energy Calculator Electric Field Potential Calculator Conductor and Sphere Capacitance Calculator Capacitors Capacitance of Parallel Plate, Cylindrical and Spherical Capacitors Calculator Electric Field Strength in Parallel Plate, Cylindrical and Spherical Capacitors Calculator Voltage (Potential Difference) of Parallel Plate, Cylindrical and Spherical Capacitors Calculator Distance Between Plates in Parallel Plate Capacitor Calculator Plate Area in Parallel Plate Capacitor Calculator Energy Stored in Charged Capacitor Calculator Energy Stored in Parallel Plate, Cylindrical and Spherical Charged Capacitors Calculator Volumetric Energy Density of Parallel Plate, Cylindrical and Spherical Capacitors Calculator Astronomy Calculators Object Weight on Other Planets Gravitational Acceleration on the Planets of the Solar System and Their Satellites Generators Random Number Generator Password Generator