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JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Vector Algebra

Let

O

be the origin. Let

O P = x \hat{i} + y \hat{j} - \hat{k}

and

O Q = - \hat{i} + 2 \hat{j} + 3 x \hat{k}, x, y \in R, x > 0

, be such that

∣ P Q ∣ = 20

and the vector

O P

is perpendicular to

O Q

. If

O R = 3 \hat{i} + z \hat{j} - 7 \hat{k}, z \in R

, is coplanar with

O P

and

O Q

, then the value of

x^{2} + y^{2} + z^{2}

is equal to:

(A)9✓ Correct
(B)2
(C)7
(D)1

Answer: (A)

Solution

Perpendicularity gives

O P \cdot O Q = - x + 2 y - 3 x = 0

, so

y = 2 x

.
Next,

P Q = - (1 + x) \hat{i} + (2 - y) \hat{j} + (3 x + 1) \hat{k}

, and

∣ P Q ∣^{2} = 20

with

y = 2 x

gives

(1 + x)^{2} + (2 - 2 x)^{2} + (3 x + 1)^{2} = 14 x^{2} + 6 = 20

so

x^{2} = 1

, and

x > 0

forces

x = 1

, hence

y = 2

.
Coplanarity of

O P = (1, 2, - 1)

,

O Q = (- 1, 2, 3)

and

O R = (3, z, - 7)

requires a vanishing scalar triple product:

1 - 1 3 22 z - 1 3 - 7 = (- 14 - 3 z) - 2 (7 - 9) - (- z - 6) = - 4 - 2 z = 0

so

z = - 2

and

x^{2} + y^{2} + z^{2} = 1 + 4 + 4 = 9

.

JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Single correct option

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