Angle Between Two Vectors Calculator
How to use a Angle Between Two Vectors Calculator?
Here are the steps to use the Angle Between Two Vectors Calculator to find the Angle Between Two Vectors.:
Step-1: Input the Components of Vector 1:
Enter the x-component of the first vector in the first input box under “Vector 1 (x₁, y₁, z₁)”.
Enter the y-component of the first vector in the second input box under “Vector 1 (x₁, y₁, z₁)”.
Enter the z-component of the first vector in the third input box under “Vector 1 (x₁, y₁, z₁)”.
Step-2: Input the Components of Vector 2:
Enter the x-component of the second vector in the first input box under “Vector 2 (x₂, y₂, z₂)”.
Enter the y-component of the second vector in the second input box under “Vector 2 (x₂, y₂, z₂)”.
Enter the z-component of the second vector in the third input box under “Vector 2 (x₂, y₂, z₂)”.
Step-3: Use Fractions if Needed:
If any of the vector components are fractions, you can enter them in the format numerator/denominator (e.g., 1/2).
Step-4: Calculate the Angle:
After entering all the vector components, click on the “Calculate Angle” button.
Step-5: View the Result:
The angle between the two vectors will be displayed in degrees in the result section below the button.
That’s it! By following these steps, you can easily find the angle between two vectors using this calculator.
How to Find the Angle Between Two Vectors?
Here are the steps to find the Angle Between Two Vectors manually:
Step-1: Understand the Vectors:
Let A=(x₁, y₁, z₁) and B=(x₂, y₂, z₂) be the two vectors.
Step-2: Compute the Dot Product:
The dot product A⋅B is calculated as:
A⋅B=x₁⋅x₂+y₁⋅y₂+z₁⋅z₂
Step-3: Compute the Magnitudes of the Vectors:
The magnitude (or length) of the vector A is:
\left|A\right|=\sqrt{x_1^2+y_1^2+z_1^2}
The magnitude (or length) of the vector B is:
\left|B\right|=\sqrt{x_2^2+y_2^2+z_2^2}
Step-4: Calculate the Cosine of the Angle:
The cosine of the angle θ between the vectors can be found using the dot product and magnitudes:
\cos \left(\theta \right)=\frac{A\cdot B}{\left|A\right|\cdot \left|B\right|}
Step-5: Solve for the angle θ:
\theta =\cos ^{-1}\left(\frac{A\cdot B}{\left|A\right|\cdot \left|B\right|}\right)
Example of the Angle Between Two Vectors.
Let’s find the angle between vectors A=(1,2,3) and B=(4,5,6):
Step-1: Understand the Vectors:
Let A=(1,2,3) and B=(4,5,6) be the two vectors.
Step-2: Compute the Dot Product:
The dot product A⋅B is calculated as:
A\cdot B=1 \cdot 4+2\cdot 5+3\cdot 6\\
A\cdot B=4+10+18\\
A \cdot B=32\\
Step-3: Compute the Magnitudes of the Vectors:
The magnitude (or length) of the vector A is:
\left|A\right|=\sqrt{1^2+2^2+3^2}\\
\left|A\right|=\sqrt{1+4+9}\\
\left|A\right|=\sqrt{14}\\
The magnitude (or length) of the vector B is:
\left|B\right|=\sqrt{4^2+5^2+6^2}\\
\left|B\right|=\sqrt{16+25+36}\\
\left|B\right|=\sqrt{77}\\
Step-4: Calculate the Cosine of the Angle:
\cos \left(\theta \right)=\frac{A\cdot B}{\left|A\right|\cdot \left|B\right|}\\
\cos \left(\theta \right)=\frac{32}{\sqrt{14}\cdot \sqrt{77}}\\
\cos \left(\theta \right)=\frac{32}{\sqrt{1078}}\\
\cos \left(\theta \right)=0.97463184\\
Step-5: Solve for the angle θ:
\theta =\cos ^{-1}\left(0.97463184\right)\\
\theta \approx 12.9^{\circ }\\
FAQs on the angle between any two vectors.
1) Can the angle between two vectors be negative?
Answer: No, the angle between two vectors is always between 0 and 180 degrees.
2) What is the angle between two identical vectors?
Answer: The angle between two identical vectors is 0 because they point in the same direction.
3) What is the angle between two orthogonal (perpendicular) vectors?
Answer: The angle between two orthogonal vectors is 90 degrees.
4) What happens if one of the vectors is a zero vector?
Answer: The angle between a zero vector and any other vector is undefined because the magnitude of the zero vector is zero, leading to a division by zero in the cosine calculation.
