Question1The points A, B and O is the origin, such that where, m, n are non-parallel vectors. M is the mid-point of AB. And E lies on AB such that AE : EB=2:1a) Find (2 marks)b) Find (2 marks)Question2O is the origin, the points A and B have position vectors and respectively. Finda) the vector (1 mark)b) (2 mark)c) the position vector of the mid-point of AB (2 mark)d) the position vector of the point C such that OABC is a parallelogram (1 mark)

Question

Question1The points A, B and O is the origin, such that where, m, n are non-parallel vectors. M is the mid-point of AB. And E lies on AB such that AE : EB=2:1a) Find (2 marks)b) Find (2 marks)Question2O is the origin, the points A and B have position vectors and respectively. Finda) the vector (1 mark)b) (2 mark)c) the position vector of the mid-point of AB (2 mark)d) the position vector of the point C such that OABC is a parallelogram (1 mark)

BlackTom AI · Accepted Answer

We will first restate the problems as clearly as possible given the text:
Question 1: The points A, B lie in a plane with O as the origin, and A and B are represented by position vectors m and n (non-parallel). M is the midpoint of AB. E lies on AB such that AE : EB = 2 : 1. a) Find ?. b) Find ?.
Question 2: O is the origin, the points A and B have position vectors a and b respectively. Find a) the vector ? b) ? c) the position vector of the midpoint of AB d) the position vector of the point C such that OABC is a parallelogram.
Since the provided text of each question includes missing target items (what to find exactly), I’ll outline the standard interpretations and show the key relationships you would use to compute each likely quantity. Where a specific symbol is missing, I annotate the typical result you would obtain for that item.
Question 1
- Let OA = m and OB = n, with O the origin. Then:
  • The midpoint M of AB has position vector OM = (OA + OB)/2 = (m + n)/2. This gives the natural answer for “M is the midpoint of AB” in vector form.
  • The point E lies on AB with AE:EB = 2:1. To locate E along the line AB, use the section formula: if E divides AB in the ratio AE:EB = 2:1, then E = (1*A + 2*B)/3 in vector form, so OE = (m + 2n)/3. This is the standard internal division result: E is closer to B since EB is smaller (1 part).
Notes and common variations you might be asked to provide:
- If the prompt asks for the vector AB, that’s simply AB = OB − OA = n − m.
- If asked for the vector OM, we already have OM = (m + n)/2.
Question 2
- Given OA = a and OB = b, with O as the origin:
  • The vector AB (from A to B) is AB = OB − OA = b − a.
  • The midpoint of AB has position vector (OA + OB)/2 = (a + b)/2, so the midpoint M has OM = (a + b)/2.
  • If OABC is a parallelogram with O and A, B already given, the fourth vertex C must satisfy OC = OA + OB (parallelogram rule: adjacent sides OA and OB sum as the diagonal endpoint). Therefore OC has position vector a + b, and the point C has position vector a + b.
Putting it together, here are the standard results you can cite:
- AB = b − a.
- Midpoint of AB: OM = (a + b)/2.
- Parallelogram vertex C: OC = a + b (so C is at position a + b).

If your original questions (a) and (b) in Question 1 or parts (a)-(d) in Question 2 specify different target quantities, substitute them with the same underlying ideas: use midpoint formulas for averages of position vectors, use section formula for internal division along AB, use AB = OB − OA for the segment vector, and use OC = OA + OB for the parallelogram completion. Always express results in terms of the given position vectors (m, n, a, b) and keep the origin O as the zero vector when computing derived vectors.

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