Question 6:
The points P, Q and R are collinear. It is given that →PQ=4a˜−2b˜ and →QR=3a˜+(1+k)b˜ , where k is a constant. Find
(a) the value of k,
(b) the ratio of PQ : QR.
Solution:
(a)
Note: If P, Q and R are collinear,→PQ=m→QR4a˜−2b˜=m[3a˜+(1+k)b˜]4a˜−2b˜=3ma˜+m(1+k)b˜Comparing vector:a˜: 4=3m m=43b˜: −2=m(1+k)−2=43(1+k)1+k=−64k=−32−1k=−52
(b)
→PQ=m→QR→PQ=43→QR→PQ→QR=43∴PQ:QR=4:3
The points P, Q and R are collinear. It is given that →PQ=4a˜−2b˜ and →QR=3a˜+(1+k)b˜ , where k is a constant. Find
(a) the value of k,
(b) the ratio of PQ : QR.
Solution:
(a)
Note: If P, Q and R are collinear,→PQ=m→QR4a˜−2b˜=m[3a˜+(1+k)b˜]4a˜−2b˜=3ma˜+m(1+k)b˜Comparing vector:a˜: 4=3m m=43b˜: −2=m(1+k)−2=43(1+k)1+k=−64k=−32−1k=−52
(b)
→PQ=m→QR→PQ=43→QR→PQ→QR=43∴PQ:QR=4:3
Question 7:
Given that x˜=3i˜+mj˜ and y˜=4i˜−3j˜ , find the values of m if the vector x˜ is parallel to the vector y˜ .
Solution:
If vector x˜ is parallel to vector y˜x˜=hy˜(3i˜+mj˜)=h(4i˜−3j˜)3i˜+mj˜=4hi˜−3hj˜Comparing vector:i˜: 3=4h h=34j˜: m=−3h m=−3(34)=−94
Given that x˜=3i˜+mj˜ and y˜=4i˜−3j˜ , find the values of m if the vector x˜ is parallel to the vector y˜ .
Solution:
If vector x˜ is parallel to vector y˜x˜=hy˜(3i˜+mj˜)=h(4i˜−3j˜)3i˜+mj˜=4hi˜−3hj˜Comparing vector:i˜: 3=4h h=34j˜: m=−3h m=−3(34)=−94