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Vector Addition in 3-D

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Visualizing Vector Addition in 3-D 

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Define vector a 

`:=`(a, `<,>`(3, 2, 0)) 

Define vector b 

 

`:=`(b, `<,>`(-4, 4, 7)) 

Compute the vector sum a + b 

 

`+`(a, b) =  

 

 

Diagram of the vector sum `+`(a, b).  The resultant is shown in black. Important:  There are two ways to draw the sum of two vectors, but the resultant doesn't change!  
It's like taking two different routes to get to the same school. 

VectorSumPlot(a, b, vectorcolors = [red, blue], show = 1) 

 

VectorSumPlot(a, b, vectorcolors = [red, blue]) 

 

 

What happens to the sum if we scale vector b by a scalar s? 

`+`(a, `*`(s, `*`(b))) =  

 

Animation of the vector sum `+`(a, sb), where s ranges from 0 to 1.   

(Click on the picture, then click play in the animation toolbar.  Try rotating the 3-D picture with the mouse!) 

 

Note that the resultant not only lengthens but also changes angle! 

 

animate(VectorSumPlot, [a, `*`(s, `*`(b)), vectorcolors = [red, blue], show = 1], s = 0 .. 1, frames = 100) 

 

Let's add a third vector, c 

`:=`(c, `<,>`(5, 0, 6)) 

Here are the vectors we're now working with 

a, b, c 

The vector sum a + b + c 

`+`(a, b, c) =  

 

 

Let's visualize this.   

 

There are six possible "routes" to the tip of the resultant, `<,>`(4, 6, 13):
a + b + ca + c + bb + a + cb + c + ac + a + bc + b + aCan you trace them all? 

VectorSumPlot(a, b, c, vectorcolors = [red, blue, green], view = [-5 .. 13, -5 .. 13, -5 .. 13])
VectorSumPlot(a, b, c, vectorcolors = [red, blue, green], view = [-5 .. 13, -5 .. 13, -5 .. 13]) 

 

One of the six routes shown in isolation. 

VectorSumPlot(a, b, c, vectorcolors = [red, blue, green], show = 1, view = [-5 .. 13, -5 .. 13, -5 .. 13])
VectorSumPlot(a, b, c, vectorcolors = [red, blue, green], show = 1, view = [-5 .. 13, -5 .. 13, -5 .. 13]) 

 

 

What effect does c have on the sum? 

 Here is an animation of the vector sum 

 

`+`(a, b, sc),  

 

where s ranges from  

0 to 3 

 

animate(VectorSumPlot, [a, b, `*`(s, `*`(c)), vectorcolors = [red, blue, green], show = 1], s = 0 .. 3, frames = 100)
animate(VectorSumPlot, [a, b, `*`(s, `*`(c)), vectorcolors = [red, blue, green], show = 1], s = 0 .. 3, frames = 100)