Let a = 2i + λ_1j + 3k, b = 4i + (3-λ_2)j + 6k and c = 3i + 6j + (λ_3 - 1)k be three vectors such that b = 2a and a is perpendicular to c. Then a possible value of (λ_1, λ_2, λ_3) is

Let a = 2i + λ_1j + 3k, b = 4i + (3-λ_2)j + 6k and c = 3i +… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 10 Jan · Shift I

Let

\mathbf{a} = 2\hat{i} + \lambda_1\hat{j} + 3\hat{k}

\mathbf{b} = 4\hat{i} + (3-\lambda_2)\hat{j} + 6\hat{k}

and

\mathbf{c} = 3\hat{i} + 6\hat{j} + (\lambda_3 - 1)\hat{k}

be three vectors such that

\mathbf{b} = 2\mathbf{a}

and

\mathbf{a}

is perpendicular to

\mathbf{c}

. Then a possible value of

(\lambda_1, \lambda_2, \lambda_3)

A.
(1, 3, 1)
B.
(1, 5, 1)
C.
$\left(-\dfrac{1}{2}, 4, 0\right)$
D.
$\left(\dfrac{1}{2}, 4, -2\right)$

Show answer and solution

Correct option: C

Solution

From b = 2a: (4, 3-λ₂, 6) = (4, 2λ₁, 6), so 3-λ₂ = 2λ₁.

From a ⊥ c: a·c = 0 → 2(3) + λ₁(6) + 3(λ₃-1) = 0 → 6 + 6λ₁ + 3λ₃ - 3 = 0 → 6λ₁ + 3λ₃ = -3 → 2λ₁ + λ₃ = -1.

For option 3: λ₁ = -1/2, λ₂ = 4, λ₃ = 0.
Check b = 2a: 3-λ₂ = 3-4 = -1 = 2(-1/2) = 2λ₁ ✓
Check perpendicularity: 2λ₁ + λ₃ = 2(-1/2) + 0 = -1 ✓

[JEE Main 2019 · 10 Jan · Shift I]

Practise Vector Algebra the way it is asked

A timed set of real JEE Main Vector Algebra questions, graded as the exam grades. No sign-up.

Start a Vector Algebra test

More: all JEE Main Vector Algebra questions · JEE Main 2019 · Mathematics chapter list

JEE Main 2019 Mathematics question on Vector Algebra