Vectors

[Zongear]

Hello, I have completed part a of this problem, but am stuck on finding the vector OX. I have calculated NM to be 1/2(a-c) but now need to work out what fraction of NM the vector NX is. Am I missing something about the geometry of the parallelogram/triangle that helps with this ratio?

 

[Dr.Peterson]

Hello, I have completed part a of this problem, but am stuck on finding the vector OX. I have calculated NM to be 1/2(a-c) but now need to work out what fraction of NM the vector NX is. Am I missing something about the geometry of the parallelogram/triangle that helps with this ratio?

View attachment 34370

Please show your work and results for (a) so we can see if anything you have done there is useful for (b) ... or wrong.

Then, try expressing OX in two different ways in terms of a and c, one using the scalar OX/OB, and the other using the scalar NX/NM (or MX/MN). You will be able to solve for those two scalars, using the fact that a and c are independent.

 

[blamocur]

Please show your work and results for (a) so we can see if anything you have done there is useful for (b) ... or wrong.

Then, try expressing OX in two different ways in terms of a and c, one using the scalar OX/OB, and the other using the scalar NX/NM (or MX/MN). You will be able to solve for those two scalars, using the fact that a and c are independent.

You can also take a closer look at the ANB triangle. Hint: the drawing is somewhat distorted. A more accurate one might help, as it often does.

 

[Zongear]

Ok here is what I have so far. I have tried both suggestions but am still getting nowhere.

 

[Dr.Peterson]

When I said

Then, try expressing OX in two different ways in terms of a and c, one using the scalar OX/OB, and the other using the scalar NX/NM (or MX/MN). You will be able to solve for those two scalars, using the fact that a and c are independent.

I had in mind defining constants representing those quantities, say x = OX/OB, and y = NX/NM. So I would write your

​

as

​

Since these are equal,

x(2a + c) = y/2(a - c)​

Collect terms with a on one side, and those with c on the other, and think about what the independence of a and c implies.

 

[Zongear]

x(2a+c)=c+a+y/2(a-c)

2ax-a-(a/2)y = c - cx - (c/2)y

a(2x - (1/2)x -1) = c(1-x-(1/2)y)

Like this?

 

[blamocur]

Ok here is what I have so far. I have tried both suggestions but am still getting nowhere.

View attachment 34384

Hint: the two lines inside ANB are both medians.

 

[Dr.Peterson]

x(2a+c)=c+a+y/2(a-c)

2ax-a-(a/2)y = c - cx - (c/2)y

a(2x - (1/2)x -1) = c(1-x-(1/2)y)

Like this?

I see one probable typo in the third line; check the details.

Now, what can you conclude from the fact that a multiple of a is equal to a multiple of c? Something must be true of those coefficients.

 

[Zongear]

I see one probable typo in the third line; check the details.

Now, what can you conclude from the fact that a multiple of a is equal to a multiple of c? Something must be true of those coefficients.

This suggests to me that either a and c are parallel, or the coefficients themselves contain vectors, both of which seem untrue.

 

[Dr.Peterson]

This suggests to me that either a and c are parallel, or the coefficients themselves contain vectors, both of which seem untrue.

No, the coefficients are scalars.

Hint: If two vectors u and v are (linearly) independent (in this case, not parallel), then the only way mu = nv is if m and n are zero.

 

[Zongear]

Ah I see - that leaves both x and y to be 2/3 and then I can find OX. Thank you for your help!

 

[Dr.Peterson]

Ah I see - that leaves both x and y to be 2/3 and then I can find OX. Thank you for your help!

Correct. And the hint about medians leads to the same conclusion geometrically.

This independence idea is a key to many vector proofs in geometry.