How to Find Perpendicular Vectors in 2 Dimensions
Recall the formula for slope., Read the components of the given vector., Calculate the slope., Recall the geometric definition of perpendicular slopes., Identify the reciprocal of the vector slope., Find the negative reciprocal., Write the new...
Step-by-Step Guide
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Step 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=ΔxΔy{\displaystyle {\text{slope}}={\frac {\Delta x}{\Delta y}}} -
Step 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 Δx{\displaystyle \Delta x}.
The second coefficient j{\displaystyle j} represents the vertical component of the vector, or the Δy{\displaystyle \Delta y}.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. , 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=ΔxΔy{\displaystyle {\text{slope}}={\frac {\Delta x}{\Delta y}}} slope=53{\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. , Two lines (including lines, line segments, or vectors) are perpendicular to each other if their slopes are negative 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: 5{\displaystyle 5} is the reciprocal of 15{\displaystyle {\frac {1}{5}}}. 23{\displaystyle {\frac {2}{3}}} is the reciprocal of 32{\displaystyle {\frac {3}{2}}}. 1{\displaystyle 1} is the reciprocal of 1{\displaystyle 1}. , After you have calculated the slope of your vector, find the reciprocal of that slope.Using the example that was started above, the vector with components (3,5){\displaystyle (3,5)} has a slope of 53{\displaystyle {\frac {5}{3}}}.
The reciprocal of 53{\displaystyle {\frac {5}{3}}} is 35{\displaystyle {\frac {3}{5}}}. , 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.In the working example, the original slope was 53{\displaystyle {\frac {5}{3}}}, so the slope of the perpendicular vector must be −35{\displaystyle
-{\frac {3}{5}}}. , 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.For the working example, the new vector will be (5,−3){\displaystyle (5,-3)}. -
Step 3: Calculate the slope.
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Step 4: Recall the geometric definition of perpendicular slopes.
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Step 5: Identify the reciprocal of the vector slope.
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Step 6: Find the negative reciprocal.
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Step 7: Write the new vector in component form.
Detailed Guide
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=ΔxΔy{\displaystyle {\text{slope}}={\frac {\Delta x}{\Delta y}}}
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 Δx{\displaystyle \Delta x}.
The second coefficient j{\displaystyle j} represents the vertical component of the vector, or the Δy{\displaystyle \Delta y}.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. , 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=ΔxΔy{\displaystyle {\text{slope}}={\frac {\Delta x}{\Delta y}}} slope=53{\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. , Two lines (including lines, line segments, or vectors) are perpendicular to each other if their slopes are negative 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: 5{\displaystyle 5} is the reciprocal of 15{\displaystyle {\frac {1}{5}}}. 23{\displaystyle {\frac {2}{3}}} is the reciprocal of 32{\displaystyle {\frac {3}{2}}}. 1{\displaystyle 1} is the reciprocal of 1{\displaystyle 1}. , After you have calculated the slope of your vector, find the reciprocal of that slope.Using the example that was started above, the vector with components (3,5){\displaystyle (3,5)} has a slope of 53{\displaystyle {\frac {5}{3}}}.
The reciprocal of 53{\displaystyle {\frac {5}{3}}} is 35{\displaystyle {\frac {3}{5}}}. , 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.In the working example, the original slope was 53{\displaystyle {\frac {5}{3}}}, so the slope of the perpendicular vector must be −35{\displaystyle
-{\frac {3}{5}}}. , 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.For the working example, the new vector will be (5,−3){\displaystyle (5,-3)}.
About the Author
Christine Brooks
Specializes in breaking down complex creative arts topics into simple steps.
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