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JEE Main 2023 · 24 Jan · Shift 2

JEE Main 2023 Mathematics question on Vector Algebra

Let α⃗=4i^+3j^+5k^\vec{\alpha} = 4\hat{i} + 3\hat{j} + 5\hat{k} and β⃗=i^+2j^−4k^\vec{\beta} = \hat{i} + 2\hat{j} - 4\hat{k}. Let β1⃗\vec{\beta_1} be parallel to α⃗\vec{\alpha} and β2⃗\vec{\beta_2} be perpendicular to α⃗\vec{\alpha}. If β⃗=β1⃗+β2⃗\vec{\beta} = \vec{\beta_1} + \vec{\beta_2}, then the value of 5β2⃗⋅(i^+j^+k^)5\vec{\beta_2} \cdot (\hat{i} + \hat{j} + \hat{k}) is
  1. A.
    6
  2. B.
    9
  3. C.
    11
  4. D.
    7
Show answer and solution

Correct option: D

Solution

β₁ is the projection of β onto α.

β₁ = (β·α/|α|²)α

β·α = 4(1) + 3(2) + 5(-4) = 4 + 6 - 20 = -10

|α|² = 16 + 9 + 25 = 50

So β₁ = (-10/50)α = (-1/5)(4î + 3ĵ + 5k̂)

β₂ = β - β₁ = (î + 2ĵ - 4k̂) + (1/5)(4î + 3ĵ + 5k̂)

β₂ = (1 + 4/5)î + (2 + 3/5)ĵ + (-4 + 1)k̂ = (9/5)î + (13/5)ĵ + (-3)k̂

5β₂ = 9î + 13ĵ - 15k̂

5β₂·(î + ĵ + k̂) = 9 + 13 - 15 = 7

[JEE Main 2023 · 24 Jan · Shift 2]

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