We need to find the value of "a", when two vectors are perpendicular.
We know a formula
If Two Line are represented bye vectors A⃗\vec {A}A = a1i⃗+a2j⃗+a3k⃗a_1 \vec {i} + a_2 \vec {j} + a_3 \vec {k}a1i+a2j+a3k
B⃗=b1i⃗+b2j⃗+b3k⃗\vec {B} = b_1 \vec {i} + b_2 \vec {j} + b_3 \vec {k}B=b1i+b2j+b3k are perpendicular
Then A⃗.B⃗=0\vec {A} . \vec {B} = 0A.B=0
It means a1b1+a2b2+a3b3=0a_1 b_1 + a_2 b_2 + a_3 b_3 = 0a1b1+a2b2+a3b3=0
The given lines are A⃗=2i⃗+aj⃗+1k⃗\vec {A} = 2 \vec {i} + a \vec {j} + 1 \vec {k}A=2i+aj+1k and B⃗=4i⃗−2j⃗−2k⃗\vec {B} = 4 \vec {i} - 2 \vec {j} - 2 \vec {k}B=4i−2j−2k are perpendicular.
So, 2(4) +a (-2) +1 (-2) = 0
2a = 6
a= 3.
Answer: a = 3
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