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Let a=−i−k,b=−i+j and c=i+2j+3k be three given vectors. If r is a vector such that r×b=c×b and r.a=0, then the value of r.b is
Numerical answer
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Correct answer: 9
Step-by-step Solution:
Analyze the given vector equation:
We are given the equation r×b=c×b.
This can be rearranged as:
r×b−c×b=0(r−c)×b=0
This equation implies that the vector (r−c) is parallel to the vector b. Two vectors are parallel if one is a scalar multiple of the other. Therefore, we can write:
r−c=λb
where λ is a scalar.
From this, we get an expression for r:
r=c+λb⋯(1)
Use the second given condition:
We are also given that r.a=0.
Substitute the expression for r from equation (1) into this condition:
(c+λb).a=0c.a+λ(b.a)=0⋯(2)
Calculate the required dot products:
The given vectors are:
a=−i−k=⟨−1,0,−1⟩b=−i+j=⟨−1,1,0⟩c=i+2j+3k=⟨1,2,3⟩
Now, we calculate c.a and b.a:
c.a=(1)(−1)+(2)(0)+(3)(−1)=−1−3=−4b.a=(−1)(−1)+(1)(0)+(0)(−1)=1+0=1
Solve for the scalar λ:
Substitute the values of the dot products back into equation (2):
−4+λ(1)=0λ=4
Calculate the final value r.b:
The question asks for the value of r.b. Using the expression for r from equation (1), we get:
r.b=(c+λb).br.b=c.b+λ(b.b)
We need to calculate c.b and b.b:
c.b=(1)(−1)+(2)(1)+(3)(0)=−1+2=1b.b=∣b∣2=(−1)2+(1)2+(0)2=1+1=2
Now substitute the values of c.b, b.b, and λ=4:
r.b=1+(4)(2)=1+8=9