Do Lines EF and GH Intersect in Vector Geometry?

[FeDeX_LaTeX]

Homework Statement

This is taken from STEP II 2003 Q5.

The position vectors of the points A, B and P with respect to an origin O are ai , bj and li + mj + nk , respectively, where a, b, and n are all non-zero. The points E, F, G and H are the midpoints of OA, BP, OB and AP, respectively. Show that the lines EF and GH intersect.

Let D be the point with position vector dk, where d is non-zero, and let S be the point of intersection of EF and GH. The point T is such that the mid-point of DT is S. Find the position vector of T and hence find d in terms of n if T lies in the plane OAB.

Homework Equations

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The Attempt at a Solution

I can't seem to make any headway with this at all, some guidance would be appreciated. There's a solution given which I don't understand:

E = a/2, F = (p+b)/2, G = b/2, H = (p+a)/2 =>
EF = a/2+x[p+b-a]/2
GH= b/2+y[p+a-b]/2
for which there is a solution EF = GH = (p+a+b)/4 when x = y = 1/2
so we have s = (d+t)/2 = (p+a+b)/4 so the position vector of T is t = (p+a+b)/2-d
the plane OAB is the x-y plane i.e. z = 0 so the component of T in the k direction is 0, so considering the vectors in the k direction we have: 0 = n/2-d <=> d = n/2

I agree with E = a/2.

I don't understand how F = (p+b)/2. If OB + BP = OP, then BP = OP - OB, so F = ½BP = ½(p-b), not ½(p+b)?

I agree with G = b/2.

I don't understand how H = (p+a)/2, for the same reason that I don't agree with F = (p+b)/2. Can anyone help?

I must be doing something very wrong because I am getting that EF and GH are identical (so intersect everywhere).

 

[HallsofIvy]

Homework Statement

This is taken from STEP II 2003 Q5.

The position vectors of the points A, B and P with respect to an origin O are ai , bj and li + mj + nk , respectively, where a, b, and n are all non-zero. The points E, F, G and H are the midpoints of OA, BP, OB and AP, respectively. Show that the lines EF and GH intersect.

Let D be the point with position vector dk, where d is non-zero, and let S be the point of intersection of EF and GH. The point T is such that the mid-point of DT is S. Find the position vector of T and hence find d in terms of n if T lies in the plane OAB.

Homework Equations

-

The Attempt at a Solution

I can't seem to make any headway with this at all, some guidance would be appreciated. There's a solution given which I don't understand:



I agree with E = a/2.

What does that mean? In the statement of the problem, above, you use bold face to mean vectors but the only "a" given is a number, not a vector.

I don't understand how F = (p+b)/2. If OB + BP = OP, then BP = OP - OB, so F = ½BP = ½(p-b), not ½(p+b)?

I agree with G = b/2.

I don't understand how H = (p+a)/2, for the same reason that I don't agree with F = (p+b)/2. Can anyone help?

I must be doing something very wrong because I am getting that EF and GH are identical (so intersect everywhere).

 

[FeDeX_LaTeX]

Apologies, I didn't make it clear -- a in non-bold text is just the constant a, whereas a or p indicate the vectors OA or OP.

 

[HallsofIvy]

Look at an example. If B= (1, 0, 0) and P= (0, 1, 0) the the point half way between them is the "average" ((1+0)/2, (0+1)/2, (0+0)/2)= (1/2, 1/2, 0). The vector form would be <1/2, 1/2, 0>= (1/2)<1, 0, 0>+ (1/2)<0, 1, 0>.

The difficulty with your reasoning is that "(1/2)BP" is NOT the vector from O to that midpoint. Yes, we can think of BP as a vector from B to P and (1/2)BP as a vector from B to the midpoint of BP. To represent (1/2)BP as a vector from O, we have to add B: <1, 0, 0>+ <-1/2, 1/2, 0>= <1/2, 1/2, 0>.