Let $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $3 \hat{i}+2 \hat{j}-\hat{k}$ and its projection on $\vec{a}$ is 19 units, then $|2 \vec{v}|^{2}$ is equal to

JEE Main · 2021 · 01 Sep 2021 · Shift 2 · Vector Algebra

Let

a = 2 \hat{i} - \hat{j} + 2 \hat{k}

and

b = \hat{i} + 2 \hat{j} - \hat{k}

. Let a vector

v

be in the plane containing

a

and

b

. If

v

is perpendicular to the vector

3 \hat{i} + 2 \hat{j} - \hat{k}

and its projection on

a

is 19 units, then

∣2 v ∣^{2}

is equal to

Answer: 1494

Write

v = λ a + μ b = (2 λ + μ) \hat{i} + (- λ + 2 μ) \hat{j} + (2 λ - μ) \hat{k}

. Perpendicularity to

3 \hat{i} + 2 \hat{j} - \hat{k}

gives

3 (2 λ + μ) + 2 (- λ + 2 μ) - (2 λ - μ) = 2 λ + 8 μ = 0

, so

λ = - 4 μ

. The projection condition gives

v \cdot a = 19 ∣ a ∣ = 57

, i.e.

9 λ - 2 μ = 57

. Substituting

λ = - 4 μ

gives

μ = - \frac{3}{2}

and

λ = 6

, so

v = \frac{21}{2} \hat{i} - 9 \hat{j} + \frac{27}{2} \hat{k}

. Then

∣2 v ∣^{2} = 4 (\frac{441}{4} + 81 + \frac{729}{4}) = 441 + 324 + 729 = 1494.

JEE Main · 2021 · 01 Sep 2021 · Shift 2 · Integer answer