Find the position vector of point R which divides the line joining two points P(2a + b) and Q(a - 3b) externally in the ratio 1 : 2. Also, show that P is the middle point of the line segment RQ.
Find the position vector of point R which divides the line joining two points P(2a + b) and Q(a - 3b) externally in the ratio 1 : 2. Also, show that P is the middle point of the line segment RQ.
It is given that OP = 2a + b, OQ = a - 3b
If a point divides the line joining point P and Q externally in the ratio m : n, then position vector of the point is m(PV of Q)−n(PV of P)m−n
It is given that point R divides a line segment joining two points P and Q externally in the ratio 1 : 2. Then on using the section formula, we get
Position vector of point R=(a−3b)×1−(2a+b)×21−2
=a−3b−4a−2b−1=−3a−5b−1=3a+5b
Now, position vector of mid-point of RQ =OQ+OR2
=(3a+5b)+(a−3b)2=4a+2b2=2a+b
Also, the position vector of point P = 2a + b
which shows that P is mid-point of line segment RQ.
It is given that OP = 2a + b, OQ = a - 3b
If a point divides the line joining point P and Q externally in the ratio m : n, then position vector of the point is m(PV of Q)−n(PV of P)m−n
It is given that point R divides a line segment joining two points P and Q externally in the ratio 1 : 2. Then on using the section formula, we get
Position vector of point R=(a−3b)×1−(2a+b)×21−2
=a−3b−4a−2b−1=−3a−5b−1=3a+5b
Now, position vector of mid-point of RQ =OQ+OR2
=(3a+5b)+(a−3b)2=4a+2b2=2a+b
Also, the position vector of point P = 2a + b
which shows that P is mid-point of line segment RQ.
externally
in the ratio 1: 2. Also, show that P is the mid point of the line
segment RQ.