Let \(\vec{p}=2 \hat{i}+3 \hat{j}+\hat{k}\) and \(\vec{q}=\hat{i}+2 \hat{j}+\hat{k}\) be two vectors. If a vector : \(\vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k})\) is perpendicular to each of the vectors ( \(\vec{p}+\vec{q}\) ) and ( \(\vec{p}-\vec{q}\) ), and \(|\vec{r}|=\sqrt{3}\), then \(|\alpha|+|\beta|+|\gamma|\) is equal to ____

JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Vector Algebra

Let

p = 2 \hat{i} + 3 \hat{j} + \hat{k}

and

q = \hat{i} + 2 \hat{j} + \hat{k}

be two vectors. If a vector :

r = (α \hat{i} + β \hat{j} + γ \hat{k})

is perpendicular to each of the vectors (

p + q

) and (

p - q

), and

∣ r ∣ = 3

, then

∣ α ∣ + ∣ β ∣ + ∣ γ ∣

is equal to ____

Answer: 3

p + q = 3 \hat{i} + 5 \hat{j} + 2 \hat{k}

and

p - q = \hat{i} + \hat{j} + 0 \hat{k}

.

Since

r

is perpendicular to both,

r

is parallel to their cross product:

r ∥ (p + q) \times (p - q) = \hat{i} 31 \hat{j} 51 \hat{k} 20 = \hat{i} (0 - 2) - \hat{j} (0 - 2) + \hat{k} (3 - 5) = - 2 \hat{i} + 2 \hat{j} - 2 \hat{k}

r = λ (- 2 \hat{i} + 2 \hat{j} - 2 \hat{k})

. Given

∣ r ∣ = 3

∣ λ ∣ 4 + 4 + 4 = ∣ λ ∣ \cdot 23 = 3 \Rightarrow ∣ λ ∣ = \frac{1}{2}

Thus

r = \pm (- \hat{i} + \hat{j} - \hat{k})

, so

(α, β, γ) = (\mp 1, \pm 1, \mp 1)

∣ α ∣ + ∣ β ∣ + ∣ γ ∣ = 1 + 1 + 1 = 3

JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Integer answer