How To Find Perpendicular Vectors In 2 Dimensions - WikiHow Life

Skip to Content

• EXPLORERandom Article

• Home
• Random

Terms of UseHow to Find Perpendicular Vectors in 2 Dimensions Explore this Article parts 1 Determining the Slope of the Original Vector 2 Calculating the Perpendicular Slope Other Sections Questions & Answers Tips and Warnings Related Articles References Co-authored by wikiHow Staff

Last Updated: July 20, 2025

X

This article was co-authored by wikiHow Staff. Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. This article has been viewed 69,454 times.

A vector is a mathematical tool for representing the direction and magnitude of some force. You may occasionally need to find a vector that is perpendicular, in two-dimensional space, to a given vector. This is a fairly simple matter of treating the vector as a line segment and finding the negative reciprocal of that line segment.

Steps

Part 1 Part 1 of 2:

Determining the Slope of the Original Vector

1. 1 Recall the formula for slope. The slope of any given line or line segment is calculated by dividing the vertical change (or the “rise”) by the horizontal change (the “run”). This can be expressed more symbolically as follows:
   • slope = Δ x Δ y {\displaystyle {\text{slope}}={\frac {\Delta x}{\Delta y}}}
2. 2 Read the components of the given vector. A vector can be written in component form as ( i , j ) {\displaystyle (i,j)} . In this form, the first coefficient i {\displaystyle i} represents the horizontal component of the vector, or the Δ x {\displaystyle \Delta x} . The second coefficient j {\displaystyle j} represents the vertical component of the vector, or the Δ y {\displaystyle \Delta y} .[1]
   • For this article, we assume that you are given the vector in its component form. If, instead, you have the vector in angle-magnitude form, you will need to calculate the components first. For help with that, see Resolve a Vector Into Components.
   Advertisement
3. 3 Calculate the slope. To find the slope, fill in the vector components into the formula for the slope. Specifically, you will divide the j {\displaystyle j} component by the i {\displaystyle i} component.
   • For example, suppose you have a vector represented as ( 3 , 5 ) {\displaystyle (3,5)} . This means that the horizontal change is 3 {\displaystyle 3} , and the vertical change is 5 {\displaystyle 5} . Find the slope:
     • slope = Δ x Δ y {\displaystyle {\text{slope}}={\frac {\Delta x}{\Delta y}}}
     • slope = 5 3 {\displaystyle {\text{slope}}={\frac {5}{3}}}
   • You could convert this result to a decimal, which would be 1.6. However, leaving it in fraction form will actually be easier for finding the perpendicular slope.
   Advertisement

Part 2 Part 2 of 2:

Calculating the Perpendicular Slope

1. 1 Recall the geometric definition of perpendicular slopes. Two lines (including lines, line segments, or vectors) are perpendicular to each other if their slopes are negative reciprocals.[2]
   • Recall that a reciprocal is the multiplicative inverse of a given number. For a fraction, this can mean just “flipping” the fraction upside down. The following are examples of some numbers and their reciprocals:
     • 5 {\displaystyle 5} is the reciprocal of 1 5 {\displaystyle {\frac {1}{5}}} .
     • 2 3 {\displaystyle {\frac {2}{3}}} is the reciprocal of 3 2 {\displaystyle {\frac {3}{2}}} .
     • 1 {\displaystyle 1} is the reciprocal of 1 {\displaystyle 1} .
2. 2 Identify the reciprocal of the vector slope. After you have calculated the slope of your vector, find the reciprocal of that slope.[3]
   • Using the example that was started above, the vector with components ( 3 , 5 ) {\displaystyle (3,5)} has a slope of 5 3 {\displaystyle {\frac {5}{3}}} .
   • The reciprocal of 5 3 {\displaystyle {\frac {5}{3}}} is 3 5 {\displaystyle {\frac {3}{5}}} .
3. 3 Find the negative reciprocal. If the slope of the original vector is positive, then the slope of the perpendicular vector will have to be negative. Conversely, if the slope of the original vector is negative, then the slope of the perpendicular vector will be positive.[4]
   • In the working example, the original slope was 5 3 {\displaystyle {\frac {5}{3}}} , so the slope of the perpendicular vector must be − 3 5 {\displaystyle -{\frac {3}{5}}} .
4. 4 Write the new vector in component form. Knowing the slope is almost the final step. You then need just to rewrite the vector in its component form, using the “rise” and “run” components.[5]
   • For the working example, the new vector will be ( 5 , − 3 ) {\displaystyle (5,-3)} .
   Advertisement

Community Q&A

Search Add New Question

• Question Suppose u and v are nonzero vectors in 3 space, prove that u and v are perpendicular to both u and v. Community Answer Did you perhaps mean the cross product, u x v? In any case, the usual way to prove perpendicular vectors is to show that their dot product is zero. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 0 Helpful 1

Ask a Question 200 characters left Include your email address to get a message when this question is answered. Submit Advertisement

Tips

• Notice that the negative sign could be placed on either the i or j part to get the same vector. The part with the negative sign is primarily a personal preference. Thanks Helpful 0 Not Helpful 0
• The steps in this article should provide a strong understanding for calculating a perpendicular vector. As a shortcut, you can generally just switch the two components and make one of them negative. Thanks Helpful 0 Not Helpful 0

Advertisement

You Might Also Like

How toDress Modestly for a Mormon Church (for Females) How toBecome Taller Naturally How toImprove Your Alexa Ranking How toOrganize a Small Closet How toSlim Your Face How toFix Your Whole Life How toKill Fruit Flies How toInsert a Rectal Suppository How toTip How toAsk the Right Questions How toFold Sheets How toIron How toWrite a Subcontractor Contract How toBuild Good Credit Advertisement

References

1. ↑ https://www.khanacademy.org/math/precalculus/vectors-precalc/component-form-of-vectors/e/converting-from-magnitude-and-direction-form-to-component-form
2. ↑ http://www.purplemath.com/modules/slope2.htm
3. ↑ http://www.purplemath.com/modules/slope2.htm
4. ↑ http://www.purplemath.com/modules/slope2.htm
5. ↑ https://www.khanacademy.org/math/precalculus/vectors-precalc/component-form-of-vectors/e/converting-from-magnitude-and-direction-form-to-component-form

About this article

Co-authored by: wikiHow Staff wikiHow Staff Writer This article was co-authored by wikiHow Staff. Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. This article has been viewed 69,454 times. 8 votes - 50% Co-authors: 8 Updated: July 20, 2025 Views: 69,454

• Print

Thanks to all authors for creating a page that has been read 69,454 times.

Did this article help you?

Yes No Advertisement Cookies make wikiHow better. By continuing to use our site, you agree to our cookie policy. Co-authored by: wikiHow Staff wikiHow Staff Writer 8 votes - 50% Click a star to vote 50% of people told us that this article helped them. Co-authors: 8 Updated: July 20, 2025 Views: 69,454

You Might Also Like

How toDress Modestly for a Mormon Church (for Females)How toBecome Taller NaturallyHow toImprove Your Alexa RankingHow toOrganize a Small Closet wikiHow Newsletter You're all set! Helpful how-tos delivered toyour inbox every week! Sign me up! By signing up you are agreeing to receive emails according to our privacy policy.

• Home
• About wikiHow
• Experts
• Contact Us
• Terms of Use
• Privacy Policy
• Do Not Sell or Share My Info
• Not Selling Info

Follow Us

--438