VQ - Add and Subtract Vectors Lesson

Math_Lesson_TopBanner.pngAdd and Subtract Vectors

In order to answer questions about vectors, we often want to add them and find the resultant vector. It is easiest to do this when the vectors are on the coordinate plane, or in component form.

Options One: Tip-To-Tail

Given two vectors, a and b on the coordinate plane. Let's find a + b.

Imagine that we picked up vector b, and put the tail of vector b to the tip of vector a.     

The resultant is the vector formed from the tail of a to the tip of b. We can see that LaTeX: a+b=\:<8,\:2>a+b=<8,2>.

two vectors ( a and b) originating from (0,0)

two vectors ( a and b) with the tip of b on the tail of a

two vectors ( a and b) with the tip of b on the tail of a and a+b originating at the tip of a and tail of b

Option 2: Parallelogram

Given two vectors, a and b on the coordinate plane.
Let's find a + b.

Be sure the tails of both vectors are in the same place. Then use the vectors to form a parallelogram.

The resultant is the indicated diagonal of the parallelogram. Again, we get that the resultant is LaTeX: a+b=<8,\:2>.a+b=<8,2>.

 

two vectors ( a and b) originating from (0,0)

two vectors ( a and b) originating from (0,0) with two other vectors to form a parallelogram

two vectors ( a and b) originating from (0,0) with two other vectors to form a parallelogram and a+b to intersect

Option 3: Component Method

Given two vectors, a and b on the coordinate plane.

Let's find a + b.

Identify the component form of each vector:

LaTeX: a=<3,\:4>\\b=<5,\:-2>a=<3,4>b=<5,−2>

 

If two vectors are in component form, then you can find the resultant by adding the horizontal components and vertical components.

LaTeX: a + b = <3 + 5, 4 + (-2)>\\= <8, 2>a+b=<3+5,4+(−2)>=<8,2>

two vectors ( a and b) originating from (0,0)

two vectors ( a and b) originating from (0,0) with a=<3,4> and b=<5,-2>

two vectors ( a and b) originating from (0,0) with a=<3,4> and b=<5,-2> with a third vector a+b=<8,2>

 

Watch this video to practice each method.

Subtracting Vectors Geometrically

Given two vectors, a and b on the coordinate plane. Let's find a - b.

Since we are subtracting vector b, we want to find the opposite of that vector.

Now, we can add a and -b using any method we'd like. Pictured is tip-to-tail. LaTeX: a-b=\left<4,\:2\right>.a−b=⟨4,2⟩.

vector a=<1,4>, vector b=<-3,2>

vector a=<1,4>, vector -b=<3,-2>

vector a=<1,4>, vector -b=<3,2>, a-b=<4,2>

 

Watch the video to see how to subtract vectors geometrically.

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