What is the value of b such that the scalar product of the vector î

1.	What is the value of b such that the scalar product of the vector î + ĵ + k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî + 2ĵ + 3k̂ is unity?1. -22. -13. 04. 1
Answer» Correct Answer - Option 4 : 1 Calculations: Parallel vector = the sum of the vectors 2î + 4ĵ - 5k̂ and bî + 2ĵ + 3k̂ ⇒ Parallel vector = (2î + 4ĵ - 5k̂) + (bî + 2ĵ + 3k̂) ⇒ Parallel vector = (2 + b)î + 6ĵ - 2k̂ Now, unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî + 2ĵ + 3k̂ = \(\rm \dfrac {(2 + b) \hat i+ 6 \hat j - 2\hat k}{\sqrt {(2 + b)^2 + 6^2 + 2^2}}\) Given, scalar product of the vector î + ĵ + k̂ with the unit vector is unity So, \(\rm \dfrac {(2 + b) \hat i+ 6 \hat j - 2\hat k}{\sqrt {(2 + b)^2 + 6^2 + 2^2}}\) . (î + ĵ + k̂) = 1 ⇒ \(\rm \dfrac {(2 + b) + 6 - 2}{\sqrt {(2 + b)^2 + 6^2 + 2^2}}\) = 1 ⇒\(\rm {(2 + b) + 6 - 2} ={\sqrt {(2 + b)^2 + 6^2 + 2^2}}\) Squaring both sides, we get ⇒ (b + 6)²= \(\rm(2 + b)^2 + 6^2+ 2^2\) ⇒ b² + 12b + 36 = b² + 4b + 44 ⇒ 8b = 8 ⇒ b = 1

What is the value of b such that the scalar product of the vector î + ĵ + k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî + 2ĵ + 3k̂ is unity?1. -22. -13. 04. 1

Answer» Correct Answer - Option 4 : 1

Calculations:

Parallel vector = the sum of the vectors 2î + 4ĵ - 5k̂ and bî + 2ĵ + 3k̂

⇒ Parallel vector = (2î + 4ĵ - 5k̂) + (bî + 2ĵ + 3k̂)

⇒ Parallel vector = (2 + b)î + 6ĵ - 2k̂

Now, unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî + 2ĵ + 3k̂ = \(\rm \dfrac {(2 + b) \hat i+ 6 \hat j - 2\hat k}{\sqrt {(2 + b)^2 + 6^2 + 2^2}}\)

Given, scalar product of the vector î + ĵ + k̂ with the unit vector is unity

So, \(\rm \dfrac {(2 + b) \hat i+ 6 \hat j - 2\hat k}{\sqrt {(2 + b)^2 + 6^2 + 2^2}}\) . (î + ĵ + k̂) = 1

⇒ \(\rm \dfrac {(2 + b) + 6 - 2}{\sqrt {(2 + b)^2 + 6^2 + 2^2}}\) = 1

⇒\(\rm {(2 + b) + 6 - 2} ={\sqrt {(2 + b)^2 + 6^2 + 2^2}}\)

Squaring both sides, we get

⇒ (b + 6)²= \(\rm(2 + b)^2 + 6^2+ 2^2\)

⇒ b² + 12b + 36 = b² + 4b + 44

⇒ 8b = 8

⇒ b = 1

What is the value of b such that the scalar product of the vector î + ĵ + k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî + 2ĵ + 3k̂ is unity?1. -22. -13. 04. 1

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