Question

Find the value of the constant s such that the scalar product of the vector i+j+k with the unit vector patallel to the sum of thw vectors 2i+4j-5k and si+2j +3k is equal to one

Answer 1

Let a→ = i + j + k,
b→ = 2i + 4j – 5k
and c→ = si + 2j + 3k.

And b→ + c→ 
= 2i + 4j – 5k + si + 2j + 3k
= (2 + s)i + 6j – 2k.

Therefore,       
|b→ + c→| = √{(2 + s)2 + (6)2 + (–2)2}

= √(4 + s2 + 4s+ 36 + 4)

= √(s2 + 4s + 44)

As per question,         
  a→.[{b→ + c→}/|b→ + c→|] = 1

Or,                  
(i + j + k).[{(2 + s)i + 6j – 2k}/√(s2+ 4s + 44)] = 1

Or,                   {(2 + s) + 6 – 2}/√(s2 + 4s + 44) = 1

Or,                   s + 6 = √(s2 + 4s + 44)

Or,                   Squaring both sides, we get

s2 + 36 + 12s = s2 + 4s+ 44

Or,                   s2 + 12s– s2 – 4s = 44 – 36

Or,                   8s= 8

=> s= 1.
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