Let w = i ^ + j ^ − 2 k ^ , and u and v be two vectors such that u × v = w and v × w = u . Let α , β , γ and t be real numbers such that u = α i ^ + β j ^ + γ k ^ , − t α + β + γ = 0 , α − tβ + γ

Question

Let w = i ^ + j ^ ​ − 2 k ^ , and u and v be two vectors such that u × v = w and v × w = u . Let α , β , γ and t be real numbers such that u = α i ^ + β j ^ ​ + γ k ^ , − t α + β + γ = 0 , α − tβ + γ = 0 , and α + β − t γ = 0 . Match each entry in List-I to the correct entry in List-II and choose the correct option. LIST - I LIST - II (P) ∣ v ∣ 2 is equal to (1) 0 (Q) If α = 3 ​ ,then γ 2 is equal to (2) 1 (R) If α = 3 ​ ,then ( β + γ ) 2 is equal to (3) 2 (S) If α = 2 ​ ,then t + 3 is equal to (4) 3 (5) 5

TestHub · Accepted Answer

u × v = w v × w = u ​ u ⊥ v v ⊥ w ​ u ⊥ w ​ ​ w = i ^ + j ^ ​ − 2 k ^ u = α i ^ + β j ^ ​ + γ k ^ ​ ​ − t 1 1 ​ 1 − t 1 ​ 1 1 − t ​ ​ = 0 u . w = 0 = α + β − 2 γ ( v . w ) × v = u × v w ∣ v ∣ 2 − 0 = w ∣ v ∣ = 1 ( P ) ​ t = − 1 or 2 t = − 1 α + β + γ = 0 t = 2 α = β = γ ​ ​ Also, u × v = w ∣ u ∣∣ v ∣ = ∣ w ∣ ⇒ ∣ u ∣ = ∣ w ∣ = 6 ​ ​ Case-1: ​ t = 2 α = β = γ α 2 + β 2 + γ 2 = 6 α = 2 ​ , − 2 ​ (S) t + 3 = 5 ​ Case-2: ​ t = − 1 α + β + γ = 0 α + β − 2 γ = 0 γ = 0 , α = − β ( Q ) α = 3 ​ , γ 2 = 0 , β = − 3 ​ ( R ) α = 3 ​ ( γ + β ) 2 = β 2 = 3 ( P ) → ( 2 ) , ( Q ) → ( 1 ) , ( R ) → ( 4 ) , ( S ) → ( 5 ) ​

Mathematics - Vector Question with Solution | TestHub

Options:

Doubts & Discussion

JEE Advanced (2028)

	LIST - I		LIST - II
(P)	$∣ v ∣^{2}$ is equal to	(1)	0
(Q)	If $α = 3$ ,then $γ^{2}$ is equal to	(2)	1
(R)	If $α = 3$ ,then $(β + γ)^{2}$ is equal to	(3)	2
(S)	If $α = 2$ ,then $t + 3$ is equal to	(4)	3
		(5)	5