Let a →=i^+j^+k^, b →=i^ and c ^=c1i^+c2j^+c3k^ Then, (i) If c1 = 1 and c2 = 2, find - Maths -

Question

Let   a   → = i ^ + j ^ + k ^ ,   b  
→ = i ^   and   c   ^ = c 1 i ^ + c 2 j ^ + c 3
k ^   Then ,
(i) If c 1 = 1 and c 2 = 2,
find c 3 which makes a , →   b  
→ and   c   → coplanar

(ii) If c 2 = −1 and c
3 = 1, show that no value of c 1 can
make a , →   b   → and   c   →
coplanar

Global Expert · Accepted Answer

i If c1=1 and c2=2, then a→=i^+j^+k^, b→=i^ and c^=i^+2j^+c3k^
We know that vectors a→, b→, c→ are coplanar iff a→ b→ c=0
It is given that a→, b→, c→ are coplanar
∴ a→ b→ c = 0⇒    11110012c3=0 ⇒10-0-1c3-o+12-0=0⇒-c3 + 2=0⇒c3=2

ii If c2=-1 and c3=1, then a→=i^+j^+k^, b→=i^ and c^=c1i^-j^+k
^We know that vectors a→, b→, c→ are coplanar iff a→ b→ c=0
For a→, b→, c→ to be coplanar:⇒a→ b→ c = 0⇒111100c1-11=0 ⇒10-0-11-0+1-1-0=0⇒-1-1=0⇒-2=0But this is never possible, whatever be the value of c1
 Thus, no vaue of c1 can make a→, b→ and c→ coplanar

Let a → = i ^ + j ^ + k ^ , b → = i ^ and c ^ = c 1 i ^ + c 2 j ^ + c 3 k ^ . Then , (i) If c 1 = 1 and c 2 = 2, find c 3 which makes a , → b → and c → coplanar. (ii) If c 2 = −1 and c 3 = 1, show that no value of c 1 can make a , → b → and c → coplanar.