Let p → = i ^ + j ^ + k ^ and r → be a variable vector such that r → ⋅ i ^ , r → ⋅ j ^ and r → ⋅ k ^ are even natural numbers. If r → ⋅ p → ≤ 20 , then the number of values of r → is

Question

Let p → = i ^ + j ^ + k ^ and r → be a variable vector such that r → ⋅ i ^ , r → ⋅ j ^ and r → ⋅ k ^ are even natural numbers. If r → ⋅ p → ≤ 20 , then the number of values of r → is, 20, 60, 75, 120

MARKS App · Accepted Answer

Let, r → = x i ^ + y j ^ + z k ^ r → ⋅ i ^ = x r → ⋅ j ^ = y r → ⋅ k ^ = z Integers (even natural numbers) r → ⋅ p → = x + y + z ≤ 2 0 Let, x = 2 k 1 , y = 2 k 2 , z = 2 k 3 ∀ k 1 , k 2 , k 3 ∈ N ⇒ k 1 + k 2 + k 3 ≤ 1 0 Now, adding a dummy variable ω , such that k 1 + k 2 + k 3 + ω = 1 0 k 1 = α + 1 k 2 = β + 1 k 3 = γ + 1 α , β , γ ∈ w α + β + γ + ω = 7 ⇒ n u m b e r o f w a ys = 7 + 4 - 1 C 4 - 1 = 10 C 3 Hence, the number of values of r → = 120

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