Question

a and b are non collinear. If c=(x - 2)a + b and d=(2x + 1) a-b are collinear, then find the value of x

Answer 1

: If l, m are collinear vectors then l = λ m ⇒ (2x +1)a - b = λ[ (x-2)a + b] -1 = λ and 2x + 1 = λ( x -2) ⇒ 2x + 1 = -1( x -2) ⇒ 3x = 2 - 1 = 1 ∴ x = 1 / 3

Answer 2

Solution :-


c⃗ =(x−2)(a⃗ +b⃗ )c→=(x−2)(a→+b→) and d⃗ =(2x+1)

(a⃗ −b⃗ )d→=(2x+1)(a→−b→)

are collinear  ⟹c⃗ ×d⃗ =0⟹c→×d→=0 

[Since angle between them is 0∘/180∘0∘/180∘]

=> (x−2)(2x+1)[(a⃗ +b⃗ )×(a⃗ −b⃗ )]=0⟹(x−2)(2x+1)[(a→+b→)×(a→−b→)]=0

Now use the fact that cross-product is distributive over addition and 

subtraction.

=> (2x2−3x−2)[(a⃗ +b⃗ )×a⃗ 

(a⃗ +b⃗ )Xb⃗ ]=0⟹(2x2−3x−2)[(a→+b→)×a→−(a→+b→)Xb→]=0

=> (2x2−3x−2)[b⃗ ×a⃗ −a⃗ ×b⃗ ]=0⟹(2x2−3x−2)

[b→×a→−a→×b→]=0

Now
 
b⃗ ×a⃗ =−a⃗ ×b⃗ b→×a→=−a→×b→

 [Reversing the  sense of taking cross-product brings a minus sign ]

=> b⃗ ×a⃗ −a⃗ ×b⃗ ≠0⟹b→×a→−a→×b→≠0 

because a⃗ a→ and b⃗ b→ are not collinear.

=> 2x2−3x−2=0⟹2x2−3x−2=0

Solving this quadratic equation you get, x=2x=2or x=−0.5x=−0.5.

===============
@GauravSaxena01