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I am watching Khan Academy Linear Algebra series and I am really getting confused when Sal writes some sort of equation for the vector, such as $v = ai + bj + ck$. I understand that vector $v$ should be of the form $(x, y, z, ...)$. Sal talked about vectors as lines... so it might be related to some definition of a line on the plane, but I am not really sure. I will appreciate your clarification!

Thanks for help!

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3 Answers 3

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The $i,j,k$ are the basis vectors $i=(1, 0, 0)$, $j=(0, 1, 0)$, $k=(0, 0, 1)$, so that $$ai+bj+ck=a(1,0,0)+b(0,1,0)+c(0,0,1),$$ which gives $$(a,0,0)+(0,b,0)+(0,0,c)=(a,b,c).$$ So this is basically the vector you refer to as $(x,y,z)$.

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  • $\begingroup$ thanks! But why Sal write like this? What is the benefit over usual (x, y, z) notation? $\endgroup$
    – YohanRoth
    Jan 11, 2016 at 8:00
  • $\begingroup$ For example, suppose you wanted to prove what $(x,y,z)+(u,v,w)$ was equal to. Well: $$(x,y,z)+(u,v,w)=ix+jy+kz + iu+iv+iw=i(x+u)+j(y+v)+k(z+w)=(x+u,y+v,z+w).$$ Ok, we intuitively know what $(x,y,z)+(u,v,w)$ is (if you've learned that somewhere), but how would go about proving it? You need to use your definitions and we define $(x,y,z)=ix+jy+kz$. Similarly we need to define what $i,j,k$ are and also what it means to write $ix$ etc. That's maths - there's no room for ambiguity ! ;-) $\endgroup$
    – pshmath0
    Jan 11, 2016 at 9:05
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These are standard base vectors:

  • $i = (1,0,0)$,
  • $j = (0,1,0)$,
  • $k = (0,0,1)$.
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They are the two equivalent ways of writing the same vector. If a vector $\vec{v} = (2,3,4) \iff \vec{v} = 2\vec{i}+3\vec{j}+4\vec{k}$.

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