Simple vector problems (hints please)

[sony]

Hi, I'm stuck here:

P1(1,2,3) and P2(2,2,2)
Q1: Find a unitvector that points from P1 to P2. - A unitvector is a vector with length 1, right? But then what...?

Q2: A point M lies on the center on the line from P1 to P2. What is the position vector to M?

The sides of a parallellogram are a=2i-j+k and b=i+j
Q3: Find two vector that make up the diagonals. - I don't have a clue, which sides are a and b? And WHY! does my crappy book insist on writing everything with "i, i and k" thus making everything more difficult to read. (In HS we had fx: AB=[2,5,1]...)

Thanks for hints!

 

[H_man]

If you subtract the coordinates of point p2 from point p1 you will have your vector $$V_1_2$$.

You are correct, any unit vector does have a magnitude of one.

You create your unit vector by dividing $$V_1_2$$ by the magnitude of the vector.

Thus it will have magnitude one but the direction of the vector.

The other questions are solved in a similar manner.

 

[sony]

Ok, thanks :)

But I'm still stuck with the last to questions

 

[H_man]

Okay, as for the second one if we take a 1-D case this will simplify things. So if we have $$X_1$$ and $$X_2$$ then the point halfway between the two is clearly $$( X_1 + X_2 )/2$$.

Now just apply this to each coordinate for the halfway point for P1 and P2.

Your book is correct to use that coordinate system and you should just get used to it. Think of i = x, j = y and k = z in your head until you get used to it.

As for the 3rd one... draw a rough picture and see if u can make sense of it.

 

[sony]

Thanks I got that right now. But I'm unsure about the last one. I'm not even sure how to sketch it... I mean, one is in 3D and on in 2D...

 

[HallsofIvy]

Q3: Find two vector that make up the diagonals. - I don't have a clue, which sides are a and b? And WHY! does my crappy book insist on writing everything with "i, i and k" thus making everything more difficult to read. (In HS we had fx: AB=[2,5,1]...)

There's often a difference between the "easy way" and the "right way". "i, j, k" is the standard way to write vectors. Many people would find [2,5,1] harder to read than 2i+ 5y+ j. I don't see any difference myself (although the "[" notation is less common that "(" or "<" for vectors).

As for "Find two vector that make up the diagonals. - I don't have a clue, which sides are a and b?", do you remember the "parallelogram rule" for vector addition? Suppose you make a parallelogram with vectors a and b as sides. Where is a+b?

 

[sony]

Ah, I remember. a+b is the diagonal. Thanks