Find the position vector of a point which cuts the line segment joining the points (5,-2) and (- 1, 4) in the ratio 1:2
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>>Find the position vector of a point R wh
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Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are
i
^
+2
j
^
−
k
^
and −
i
^
+
j
^
+
k
^
respectively, in the ratio 2:1,
(i) internally
(ii) externally
Medium
Solution
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The position vector of point R dividing the line segment joining two points P and Q in the ratio m:n is given by:
i. Internally:
m+n
m
Q
+n
P
ii. Externally:
m−n
m
Q
−n
P
Position vectors of P and Q are given as:
OP
=
i
^
+2
j
^
−
k
^
and
OQ
=−
i
^
+
j
^
+
k
^
(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2:1 is given by,
OR
=
2+1
2(−
i
^
+
j
^
+
k
^
)+1(
i
^
+2
j
^
−
k
^
)
=
3
−2
i
^
+2
j
^
+
k
^
+(
i
^
+2
j
^
−
k
^
)
=
3
−
i
^
+4
j
^
+
k
^
=−
3
1
i
^
+
3
4
j
^
+
3
1
k
^
(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2:1 is given by,
OR
=
2−1
2(−
i
^
+
j
^
+
k
^
)−1(
i
^
+2
j
^
−
k
^
)
=(−2
i
^
+2
j
^
+2
k
^
)−(
i
^
+2
j
^
−
k
^
)
=−3
i
^
+3
k
^