vector addition in 3 dimensional space

[wuti]

This is a vector question seems not difficult but I am having issues.

In R³ space, question is to add two vectors a, b (a,b ∈ R³). a,b,c are vectors that a=[t,0,0], b=[0,t,1], and c= a+b=[t,t,1].

I plotted the chart using software below and you can see vector c is the blue line in the direction between x axis and y axis with a height of 1.

However, in terms of vector addition principles, when adding two vectors a and b, you first moving vector b to the starting point of a (which is the origin in our case) and then add together. using this logic, a+b should equal to vector d (orange line in chart). I know that from parametric form c=[t,t,1] should be the blue line in the chart but what is wrong with the orange line using vector addition principles? thanks!

 

[Zermelo]

I don't know what class you are taking, but in Linear Algebra, R^3 vectors "start from the origin". Also, I don't understand how did you corelate a line and a vector, those are different objects. Maybe try out a different software, and refresh your knowledge about vectors. For example, a vector v = (1,1,1) is a vector who's tip is in the point x = 1, y = 1, z = 1, and it's origin is the point (0,0,0).

 

[wuti]

Thanks so much Zermelo! you are correct that b is not a vector in R^3 but a translation of a vector that is b=b1+b2=[0,t,0]+[0,0,1], b2 is a translation here. In order to calc a+b, you have to calc a+b1 and then translation by b2. a+b1=[t,0,0]+[0,t,0]=[t,t,0], which is d in the chart. then you translation d by b2 [0,0,1], so you have [t,t,0]+[0,0,1]=[t,t,1] which gives you c in the chart. My bad confused by vector and vector translation :/ thanks again for your help much appreciated!!

 

[Zermelo]

No problem
Just a quick tip, if I were you, I wouldn’t worry myself with the vector translations too much, the definition of vector addition is (x,y,z) + (a,b,c) = (x+a, y+b, z+ c). Of course you should have a visual intuition, but you should build the intuition in R^2 space, where adding two vectors means putting them together at the same origin point ( (0,0) ) and computing the diagonal of the parallelogram they span out. That is basically the same as “putting one vector’s beginning at the other’s end”. Once you grasp this firmly, R^n shouldn’t be a problem. Also, check out 3blue1brown on youtube, and his series on Linear Algebra. Cheers