4.6.3 Vector Short Questions (Question 6 & 7)


Question 6:
The points P, Q and R are collinear. It is given that   P Q = 4 a ˜ 2 b ˜  and   Q R = 3 a ˜ + ( 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 QR 4 a ˜ 2 b ˜ =m[ 3 a ˜ +( 1+k ) b ˜ ] 4 a ˜ 2 b ˜ =3m a ˜ +m( 1+k ) b ˜ Comparing vector: a ˜ : 4=3m         m= 4 3 b ˜ : 2=m( 1+k ) 2= 4 3 ( 1+k ) 1+k= 6 4 k= 3 2 1 k= 5 2

(b)
P Q = m Q R P Q = 4 3 Q R P Q Q R = 4 3 P Q : Q R = 4 : 3



Question 7:
Given that x ˜ = 3 i ˜ + m j ˜ and   y ˜ = 4 i ˜ 3 j ˜ , 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 ˜ =h y ˜ ( 3 i ˜ +m j ˜ )=h( 4 i ˜ 3 j ˜ ) 3 i ˜ +m j ˜ =4h i ˜ 3h j ˜ Comparing vector: i ˜ :  3=4h         h= 3 4 j ˜ :  m=3h         m=3( 3 4 )= 9 4

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