Difference between revisions of "Find Perpendicular Vectors in 2 Dimensions"
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== Steps == | == Steps == | ||
===Determining the Slope of the Original Vector=== | ===Determining the Slope of the Original Vector=== | ||
| − | #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:<ref>http://mathforum.org/cgraph/cslope/calculate.html</ref> | + | #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:<ref name="rf1">http://mathforum.org/cgraph/cslope/calculate.html</ref> |
#*<math>\text{slope}=\frac{\Delta x}{\Delta y}</math> | #*<math>\text{slope}=\frac{\Delta x}{\Delta y}</math> | ||
| − | #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>.<ref>https://www.khanacademy.org/math/precalculus/vectors-precalc/component-form-of-vectors/e/converting-from-magnitude-and-direction-form-to-component-form</ref> | + | #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>.<ref name="rf2">https://www.khanacademy.org/math/precalculus/vectors-precalc/component-form-of-vectors/e/converting-from-magnitude-and-direction-form-to-component-form</ref> |
#*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]]. | #*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]]. | ||
| − | #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.<ref | + | #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.<ref name="rf1" /> |
#*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: | #*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{\Delta x}{\Delta y}</math> | ||
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#*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. | #*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=== | ===Calculating the Perpendicular Slope=== | ||
| − | #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.<ref>http://www.purplemath.com/modules/slope2.htm</ref> | + | #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.<ref name="rf3">http://www.purplemath.com/modules/slope2.htm</ref> |
#*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: | #*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>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>\frac{2}{3}</math> is the reciprocal of <math>\frac{3}{2}</math>. | ||
#**<math>1</math> is the reciprocal of <math>1</math>. | #**<math>1</math> is the reciprocal of <math>1</math>. | ||
| − | #Identify the reciprocal of the vector slope. After you have calculated the slope of your vector, find the reciprocal of that slope.<ref | + | #Identify the reciprocal of the vector slope. After you have calculated the slope of your vector, find the reciprocal of that slope.<ref name="rf3" /> |
#*Using the example that was started above, the vector with components <math>(3,5)</math> has a slope of <math>\frac{5}{3}</math>. | #*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>. | #*The reciprocal of <math>\frac{5}{3}</math> is <math>\frac{3}{5}</math>. | ||
| − | #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.<ref | + | #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.<ref name="rf3" /> |
#*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>. | #*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>. | ||
| − | #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.<ref | + | #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.<ref name="rf2" /> |
#*For the working example, the new vector will be <math>(5,-3)</math>. | #*For the working example, the new vector will be <math>(5,-3)</math>. | ||
== Tips == | == Tips == | ||
