In terms of a and b:
a) Vector PQ = a(2/3) + b(1/3)
b) Vector OT = a(2/3) - b(1/3)
a). In a triangle, if T divides side PQ into segments PT and TQ in the ratio 2:1, the position vector of T, denoted as vector PT, can be expressed as (2/3) × vector OP + (1/3) × vector OQ. This ratio corresponds to the weights applied to the vectors OP and OQ. The weights sum up to 1, reflecting the fact that T is a point on the line segment PQ.
Vector PQ is a combination of vectors OP and OQ, where the weights are determined by the ratio in which T divides PQ.
b). To find the position vector of point T (vector OT), the weights applied to vector OP and OQ are determined by the ratio in which T divides PQ. Since PT:TQ = 2:1, vector OT is expressed as (2/3) × vector OP - (1/3) × vector OQ. The positive weight for OP indicates that T is closer to P than to Q, and the negative weight for OQ reflects T being closer to P.
Vector OT is a linear combination of vectors OP and OQ, with weights determined by the ratio in which T divides PQ.
In summary, for a triangle OPQ with T dividing PQ in a 2:1 ratio, the expression for vector PQ is a(2/3) + b(1/3), and the expression for vector OT is a(2/3) - b(1/3).