Mathematics · Linear Algebra
Angle Between Vectors Calculator
Find the angle between two three-dimensional vectors using their dot product.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Dot product = 0; magnitudes = 1 and 1.
- cos θ = 0 ÷ (1×1) = 0.
- θ = arccos(0) = 90°.
Understand Angle between vectors
One idea, three depths
Choose how deeply to explain Angle between vectors
Angle between vectors: Find the angle between two three-dimensional vectors using their dot product.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Angle between vectors to answer this question: find the angle between two three-dimensional vectors using their dot product? Enter Vector A: x, Vector A: y, Vector A: z, and 3 other inputs; the calculator shows Angle in degrees. For example: Perpendicular vectors (1,0,0) and (0,1,0) have dot product 0 and angle 90°. The answer tells you Angle in degrees.
Age 15Explain it to a 15-year-oldConnect it to the formula
Normalising the dot product removes vector length, leaving the cosine of their directional separation. The rule is θ = arccos[(a·b)/(‖a‖‖b‖)]. Its input values are Vector A: x, Vector A: y, Vector A: z, Vector B: x, Vector B: y, Vector B: z, and the main result is Angle in degrees. For example: Perpendicular vectors (1,0,0) and (0,1,0) have dot product 0 and angle 90°.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated angle between vectors relation over the valid real-number domain stated below. The implemented relation is θ = arccos[(a·b)/(‖a‖‖b‖)], evaluated from Vector A: x, Vector A: y, Vector A: z, Vector B: x, Vector B: y, Vector B: z to produce Angle in degrees. Normalising the dot product removes vector length, leaving the cosine of their directional separation. The angle is undefined if either vector has zero magnitude.
Inputs and valid domain
- Vector A: x must be a finite real number.
- Vector A: y must be a finite real number.
- Vector A: z must be a finite real number.
- Vector B: x must be a finite real number.
- Vector B: y must be a finite real number.
- Vector B: z must be a finite real number.
Important boundary: The angle is undefined if either vector has zero magnitude.
The formula
θ = arccos[(a·b)/(‖a‖‖b‖)]
How the calculator works through it
It substitutes Vector A: x, Vector A: y, Vector A: z, Vector B: x, Vector B: y, Vector B: z into the formula and exposes every numerical step above. The main output is Angle in degrees, accompanied by Angle in radians, Cosine of angle.
Read the result correctly
The Angle in degrees is the direct answer to “find the angle between two three-dimensional vectors using their dot product.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
Perpendicular vectors (1,0,0) and (0,1,0) have dot product 0 and angle 90°.
Where this model stops being reliable
The angle is undefined if either vector has zero magnitude.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Angle between vectors works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Angle between vectors uses θ = arccos[(a·b)/(‖a‖‖b‖)]. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Vectors and components
Component notation helps you follow how Angle between vectors combines directional or indexed values.
Review this foundation about 6 min
Optional enrichment
- Matrices and linear transformations
Matrices place Angle between vectors inside the wider language of linear systems and transformations.
Review this foundation about 7 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Vector A: x, Vector A: y, Vector A: z, Vector B: x, Vector B: y, Vector B: z.
- Evaluate the principal relationship: θ = arccos[(a·b)/(‖a‖‖b‖)].
- Return Angle in degrees and check the domain conditions described above.
Python
from math import *
def vector_angle(a1, b1, c1, a2, b2, c2) -> float:
return ((180.0 / pi) * acos(((((a1 * a2) + (b1 * b2)) + (c1 * c2)) / (sqrt((((a1 * a1) + (b1 * b1)) + (c1 * c1))) * sqrt((((a2 * a2) + (b2 * b2)) + (c2 * c2)))))))
assert abs(vector_angle(1, 0, 0, 0, 1, 0) - 90) < 1e-6 * max(1.0, abs(90))
C
#include <assert.h>
#include <math.h>
double vector_angle(double a1, double b1, double c1, double a2, double b2, double c2) {
return ((180.0 / 3.141592653589793) * acos(((((a1 * a2) + (b1 * b2)) + (c1 * c2)) / (sqrt((((a1 * a1) + (b1 * b1)) + (c1 * c1))) * sqrt((((a2 * a2) + (b2 * b2)) + (c2 * c2)))))));
}
int main(void) {
const double expected = 90;
const double actual = vector_angle(1, 0, 0, 0, 1, 0);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double vector_angle(double a1, double b1, double c1, double a2, double b2, double c2) {
return ((180.0 / std::numbers::pi) * std::acos(((((a1 * a2) + (b1 * b2)) + (c1 * c2)) / (std::sqrt((((a1 * a1) + (b1 * b1)) + (c1 * c1))) * std::sqrt((((a2 * a2) + (b2 * b2)) + (c2 * c2)))))));
}
int main() {
constexpr double expected = 90;
const double actual = vector_angle(1, 0, 0, 0, 1, 0);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double vector_angle(double a1, double b1, double c1, double a2, double b2, double c2)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern acos
global vector_angle
section .text
vector_angle:
push rbp
mov rbp, rsp
sub rsp, 240
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd [rbp-32], xmm3
movsd [rbp-40], xmm4
movsd [rbp-48], xmm5
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-72], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-80], xmm0
movsd xmm0, [rbp-72]
divsd xmm0, [rbp-80]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-32]
movsd [rbp-120], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-40]
movsd [rbp-128], xmm0
movsd xmm0, [rbp-120]
addsd xmm0, [rbp-128]
movsd [rbp-112], xmm0
movsd xmm0, [rbp-24]
mulsd xmm0, [rbp-48]
movsd [rbp-136], xmm0
movsd xmm0, [rbp-112]
addsd xmm0, [rbp-136]
movsd [rbp-104], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-8]
movsd [rbp-176], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-16]
movsd [rbp-184], xmm0
movsd xmm0, [rbp-176]
addsd xmm0, [rbp-184]
movsd [rbp-168], xmm0
movsd xmm0, [rbp-24]
mulsd xmm0, [rbp-24]
movsd [rbp-192], xmm0
movsd xmm0, [rbp-168]
addsd xmm0, [rbp-192]
movsd [rbp-160], xmm0
sqrtsd xmm0, [rbp-160]
movsd [rbp-152], xmm0
movsd xmm0, [rbp-32]
mulsd xmm0, [rbp-32]
movsd [rbp-224], xmm0
movsd xmm0, [rbp-40]
mulsd xmm0, [rbp-40]
movsd [rbp-232], xmm0
movsd xmm0, [rbp-224]
addsd xmm0, [rbp-232]
movsd [rbp-216], xmm0
movsd xmm0, [rbp-48]
mulsd xmm0, [rbp-48]
movsd [rbp-240], xmm0
movsd xmm0, [rbp-216]
addsd xmm0, [rbp-240]
movsd [rbp-208], xmm0
sqrtsd xmm0, [rbp-208]
movsd [rbp-200], xmm0
movsd xmm0, [rbp-152]
mulsd xmm0, [rbp-200]
movsd [rbp-144], xmm0
movsd xmm0, [rbp-104]
divsd xmm0, [rbp-144]
movsd [rbp-96], xmm0
movsd xmm0, [rbp-96]
call acos wrt ..plt
movsd [rbp-88], xmm0
movsd xmm0, [rbp-64]
mulsd xmm0, [rbp-88]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-56]
leave
ret
MATLAB
function result = vector_angle(a1, b1, c1, a2, b2, c2)
result = ((180.0 / pi) * acos(((((a1 * a2) + (b1 * b2)) + (c1 * c2)) / (sqrt((((a1 * a1) + (b1 * b1)) + (c1 * c1))) * sqrt((((a2 * a2) + (b2 * b2)) + (c2 * c2)))))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a1_, b1_, c1_, a2_, b2_, c2_] := ((180.0 / Pi) * ArcCos[((((a1 * a2) + (b1 * b2)) + (c1 * c2)) / (Sqrt[(((a1 * a1) + (b1 * b1)) + (c1 * c1))] * Sqrt[(((a2 * a2) + (b2 * b2)) + (c2 * c2))]))]);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Angle Between Vectors Calculator. MW SysArc Tools. https://math.mwsysarc.com/linear-algebra/angle-between-vectors-calculator
MLA 9
MW SysArc. “Angle Between Vectors Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/linear-algebra/angle-between-vectors-calculator. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Angle Between Vectors Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/linear-algebra/angle-between-vectors-calculator.
Harvard
MW SysArc (2026) ‘Angle Between Vectors Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/linear-algebra/angle-between-vectors-calculator (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_vector_angle_2026,
author = {{MW SysArc}},
title = {Angle Between Vectors Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/linear-algebra/angle-between-vectors-calculator},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Angle Between Vectors Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/linear-algebra/angle-between-vectors-calculator
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Angle between vectors do?
Find the angle between two three-dimensional vectors using their dot product.
How does the Angle between vectors work?
The calculator applies θ = arccos[(a·b)/(‖a‖‖b‖)]. Normalising the dot product removes vector length, leaving the cosine of their directional separation.
What can I learn from the Angle between vectors?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .