Question

# The scalar product of the vector →a=^i+^j+^k with a unit vector along the sum of vectors →b=2^i+4^j−5^k and →c=λ^i+2^j+3^k is equal to one. Then the value of λ is

A
1
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B
0
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C
1
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D
2
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Solution

## The correct option is C 1→a=^i+^j+^k →b=2^i+4^j−5^k →c=λ^i+2^j+3^k So, →b+→c=(2+λ)^i+6^j−2^k Unit vector along →b+→c=(2+λ)^i+6^j−2^k√(2+λ)2+36+4 Given that dot product of →a with the unit vector along the →b+→c is equal to 1. So, ((2+λ)^i+6^j−2^k√(2+λ)2+40)⋅(^i+^j+^k)=1 ⇒(2+λ)+6−2√(2+λ)2+40=1 ⇒(2+λ)+4=√(2+λ)2+40 Squaring both sides, ⇒(2+λ)2+16+8(2+λ)=(2+λ)2+40 ⇒32+8λ=40 ⇒8λ=8 ⇒λ=1

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