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The angle between two vectors is a fundamental concept in mathematics and physics that can provide important insights into the relationship between these vectors. Whether you are studying vectors in school or working on a complex problem that involves direction and magnitude, knowing how to find the angle between two vectors is crucial. This guide aims to provide a step-by-step explanation of the process, including the use of mathematical formulas and vector properties. By understanding this topic, you will be able to solve various vector-related problems and gain a deeper understanding of vector analysis.
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This article has been viewed 231,294 times.
If you are a mathematician or a graphics programmer, you will probably have to find the angle between two given vectors. In this article, the wikiHow will show you how to do just that.
Steps
Find the angle between two vectors
![Image titled Find the Angle Between Two Vectors Step 01](https://www.wikihow.com/images_en/thumb/b/ba/Find-the-Angle-Between-Two-Vectors-Step-01.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-01.jpg)
- Example: Two-dimensional vector u→{displaystyle {overrightarrow {u}}} = (2,2) and two-dimensional vector v→{displaystyle {overrightarrow {v}}} = (0,3). They can also be written as u→{displaystyle {overrightarrow {u}}} = 2 i + 2 j and v→{displaystyle {overrightarrow {v}}} = 0 i + 3 j = 3 j .
- Although two-dimensional vectors are used in this article’s example, the following instructions can be applied to vectors of any number of dimensions.
![Image titled Find the Angle Between Two Vectors Step 02](https://www.wikihow.com/images_en/thumb/4/49/Find-the-Angle-Between-Two-Vectors-Step-02.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-02.jpg)
- cosθ = (u→{displaystyle {overrightarrow {u}}} •v→{displaystyle {overrightarrow {v}}} ) / ( ||u→{displaystyle {overrightarrow {u}}}||||v→{displaystyle {overrightarrow {v}}}|| )
- ||u→{displaystyle {overrightarrow {u}}}|| means “the length of the vector u→{displaystyle {overrightarrow {u}}} “.
- u→{displaystyle {overrightarrow {u}}} • v→{displaystyle {overrightarrow {v}}} is the dot product of two vectors – this will be explained below.
![Image titled Find the Angle Between Two Vectors Step 03](https://www.wikihow.com/images_en/thumb/e/e7/Find-the-Angle-Between-Two-Vectors-Step-03.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-03.jpg)
- || u ||2 = u 12 + u 22 . If the vector has more than two components, we just keep adding +u 32 + u 42 + …
- Therefore, for a two-dimensional vector, || u || = √(u 12 + u 22 ) .
- In this example, ||u→{displaystyle {overrightarrow {u}}}|| = √(2 2 + 2 2 ) = √(8) = 2√2 . ||v→{displaystyle {overrightarrow {v}}}|| = √(0 2 + 3 2 ) = √(9) = 3 .
![Image titled Find the Angle Between Two Vectors Step 04](https://www.wikihow.com/images_en/thumb/a/a0/Find-the-Angle-Between-Two-Vectors-Step-04.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-04.jpg)
- With a graphics program, refer to Tips before reading on.
- In math u→{displaystyle {overrightarrow {u}}} •v→{displaystyle {overrightarrow {v}}} = u 1 v 1 + u 2 v 2 , where, u = (u 1 , u 2 ). If the vector has more than two components, you just add + u 3 v 3 + u 4 v 4 …
- In this example, u→{displaystyle {overrightarrow {u}}} • v→{displaystyle {overrightarrow {v}}} = u 1 v 1 + u 2 v 2 = (2)(0) + (2)(3) = 0 + 6 = 6 . This is the dot product of the vector u→{displaystyle {overrightarrow {u}}} and vector v→{displaystyle {overrightarrow {v}}} .
![Image titled Find the Angle Between Two Vectors Step 05](https://www.wikihow.com/images_en/thumb/8/8f/Find-the-Angle-Between-Two-Vectors-Step-05.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-05.jpg)
- In our example, cosθ = 6 / ( 2√2 * 3 ) = 1 / √2 = √2 / 2.
![Image titled Find the Angle Between Two Vectors Step 06](https://www.wikihow.com/images_en/thumb/e/ee/Find-the-Angle-Between-Two-Vectors-Step-06.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-06.jpg)
- In the example, cosθ = √2 / 2. Type “arccos(√2 / 2)” into the calculator to find the angle. Or, you can find the angle θ on the unit circle, where cosθ = √2 / 2. It is true for θ = π / 4 or 45º .
- Combining everything, the final formula is: angle θ = arccosine(( u→{displaystyle {overrightarrow {u}}} • v→{displaystyle {overrightarrow {v}}} ) / ( ||u→{displaystyle {overrightarrow {u}}}||||v→{displaystyle {overrightarrow {v}}}|| ))
Determine the angle formula
![Image titled Become a Cplege Professor Step 17](https://www.wikihow.com/images_en/thumb/f/f5/Become-a-Cplege-Professor-Step-17.jpg/v4-728px-Become-a-Cplege-Professor-Step-17.jpg)
- The examples below use two-dimensional vectors because they are easiest to understand and simple. Vectors of three dimensions or more have properties defined by an almost similar general formula.
![Image titled Find the Angle Between Two Vectors Step 08](https://www.wikihow.com/images_en/thumb/b/b5/Find-the-Angle-Between-Two-Vectors-Step-08.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-08.jpg)
![Image titled Find the Angle Between Two Vectors Step 09](https://www.wikihow.com/images_en/thumb/2/23/Find-the-Angle-Between-Two-Vectors-Step-09.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-09.jpg)
![Image titled Find the Angle Between Two Vectors Step 10](https://www.wikihow.com/images_en/thumb/a/a5/Find-the-Angle-Between-Two-Vectors-Step-10-Version-2.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-10-Version-2.jpg)
- || (a – b) ||2 = || a ||2 + || b ||2 – 2 || a |||| b ||cos (θ)
![Image titled Find the Angle Between Two Vectors Step 11](https://www.wikihow.com/images_en/thumb/c/c3/Find-the-Angle-Between-Two-Vectors-Step-11-Version-2.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-11-Version-2.jpg)
- ( a→{displaystyle {overrightarrow {a}}} – b→{displaystyle {overrightarrow {b}}} ) • ( a→{displaystyle {overrightarrow {a}}} – b→{displaystyle {overrightarrow {b}}} ) = a→{displaystyle {overrightarrow {a}}} • a→{displaystyle {overrightarrow {a}}} + b→{displaystyle {overrightarrow {b}}} • b→{displaystyle {overrightarrow {b}}} – 2 || a |||| b ||cos (θ)
![Image titled Find the Angle Between Two Vectors Step 12](https://www.wikihow.com/images_en/thumb/d/de/Find-the-Angle-Between-Two-Vectors-Step-12-Version-2.jpg/v4-728px-Find-the-Angle-Between-Two-Vectors-Step-12-Version-2.jpg)
- a→{displaystyle {overrightarrow {a}}} • a→{displaystyle {overrightarrow {a}}} – a→{displaystyle {overrightarrow {a}}} • b→{displaystyle {overrightarrow {b}}} – b→{displaystyle {overrightarrow {b}}} • a→{displaystyle {overrightarrow {a}}} + b→{displaystyle {overrightarrow {b}}} • b→{displaystyle {overrightarrow {b}}} = a→{displaystyle {overrightarrow {a}}} • a→{displaystyle {overrightarrow {a}}} + b→{displaystyle {overrightarrow {b}}} • b→{displaystyle {overrightarrow {b}}} – 2 || a |||| b ||cos (θ)
- – a→{displaystyle {overrightarrow {a}}} • b→{displaystyle {overrightarrow {b}}} – b→{displaystyle {overrightarrow {b}}} • a→{displaystyle {overrightarrow {a}}} = -2 || a |||| b ||cos (θ)
- -2( a→{displaystyle {overrightarrow {a}}} • b→{displaystyle {overrightarrow {b}}} ) = -2 || a |||| b ||cos (θ)
- a→{displaystyle {overrightarrow {a}}} • b→{displaystyle {overrightarrow {b}}} = || a |||| b ||cos (θ)
Advice
- To swap values and solve problems quickly, use this formula for any pair of two-dimensional vectors: cosθ = (u 1 • v 1 + u 2 • v 2 ) / (√(u 12 •) u 22 ) • √(v 12 • v 22 )).
- If you’re working with computer graphics software, you’ll most likely only have to care about the dimensions of the vectors and not their length. Use these steps to shorten the equation and speed up your program: [9] XResearch Source[10] XResearch Source
- Normalize each vector so that they have a length of 1. To do so, divide each component of the vector by its length.
- Take the dot product of the normalized vector instead of the original vector.
- Since the vector has a length of 1, we can remove the length element from the equation. Finally, the angle equation we get is arccos( u→{displaystyle {overrightarrow {u}}} • v→{displaystyle {overrightarrow {v}}} ).
- Based on the cosine formula, we can quickly determine whether an angle is an acute or obtuse angle. Start with cosθ = ( u→{displaystyle {overrightarrow {u}}} • v→{displaystyle {overrightarrow {v}}} ) / ( ||u→{displaystyle {overrightarrow {u}}}||||v→{displaystyle {overrightarrow {v}}}|| ):
- The left and right sides of the equation must have the same sign (positive or negative).
- Since the length is always positive, cosθ must have the same sign as the dot product.
- Therefore, if the dot product is positive, cosθ is also positive. We’re in the first quadrant of the unit circle, with θ < π / 2 or 90º. The required angle is an acute angle.
- If the dot product is negative, cosθ is negative. We’re in the second quadrant of the unit circle, with π / 2 < θ ≤ π or 90º < θ ≤ 180º. It’s an obtuse angle.
This article is co-authored by a team of editors and trained researchers who confirm the accuracy and completeness of the article.
The wikiHow Content Management team carefully monitors the work of editors to ensure that every article is up to a high standard of quality.
There are 9 references cited in this article that you can view at the bottom of the page.
This article has been viewed 231,294 times.
If you are a mathematician or a graphics programmer, you will probably have to find the angle between two given vectors. In this article, the wikiHow will show you how to do just that.
In conclusion, finding the angle between two vectors is a fundamental concept in vector algebra that is used in various fields such as physics, engineering, and computer graphics. By understanding the concept of dot product and the properties of vectors, one can easily calculate the angle between two vectors. The process involves finding the dot product of the two vectors, calculating the magnitudes of the vectors, and using trigonometric functions to determine the angle. It is important to note that the angle between two vectors can range from 0 degrees (when the vectors are in the same direction) to 180 degrees (when the vectors are in opposite directions). Additionally, the angle between two vectors can provide valuable information on their relationship, whether they are orthogonal, parallel, or in an arbitrary orientation. Overall, knowing how to find the angle between two vectors is a vital skill that can enhance problem-solving abilities and deepen one’s understanding of vector concepts.
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