Find Perpendicular Vectors in 2 Dimensions

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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

Determining the Slope of the Original Vector

  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:[1]
    • <math>\text{slope}=\frac{\Delta x}{\Delta y}</math>
  2. Read the components of the given vector. A vector can be written in component form as <math>(i,j)</math>. In this form, the first coefficient <math>i</math> represents the horizontal component of the vector, or the <math>\Delta x</math>. The second coefficient <math>j</math> represents the vertical component of the vector, or the <math>\Delta y</math>.[2]
    • 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.
  3. Calculate the slope. To find the slope, fill in the vector components into the formula for the slope. Specifically, you will divide the <math>j</math> component by the <math>i</math> component.[3]
    • For example, suppose you have a vector represented as <math>(3,5)</math>. This means that the horizontal change is <math>3</math>, and the vertical change is <math>5</math>. Find the slope:
      • <math>\text{slope}=\frac{\Delta x}{\Delta y}</math>
      • <math>\text{slope}=\frac{5}{3}</math>
    • 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.

Calculating the Perpendicular Slope

  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.[4]
    • 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:
      • <math>5</math> is the reciprocal of <math>\frac{1}{5}</math>.
      • <math>\frac{2}{3}</math> is the reciprocal of <math>\frac{3}{2}</math>.
      • <math>1</math> is the reciprocal of <math>1</math>.
  2. Identify the reciprocal of the vector slope. After you have calculated the slope of your vector, find the reciprocal of that slope.[5]
    • Using the example that was started above, the vector with components <math>(3,5)</math> has a slope of <math>\frac{5}{3}</math>.
    • The reciprocal of <math>\frac{5}{3}</math> is <math>\frac{3}{5}</math>.
  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.[6]
    • In the working example, the original slope was <math>\frac{5}{3}</math>, so the slope of the perpendicular vector must be <math>-\frac{3}{5}</math>.
  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.[7]
    • For the working example, the new vector will be <math>(5,-3)</math>.

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.
  • 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.

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Sources and Citations