If r-3-2-5k- a-2-k- b-3-2k- and c-2-3k- such that r-a-b-vc- Then which of the following is true

We have, r=3î+2ĵ 5k̂⋯(i) and r=λa+μb+vc ⇒r=λ(2î ĵ+k̂)+μ(î+3ĵ 2k̂)+v( 2î+ĵ 3k̂) ⇒r= î(2λ+μ 2v)+ĵ( λ+3μ+v)+k̂(λ 2μ 3v)⋯(ii) Comparing like vectors, we get 2λ+μ 2v ...

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Byju's Answer

Standard X

Mathematics

Section Formula

If r=3î+2ĵ-5k...

Question

If $\to r = 3^i + 2^j - 5^k, \to a = 2^i -^j +^k, \to b =^i + 3^j - 2^k$ and $\to c = - 2^i +^j - 3^k$ such that $\to r = λ \to a + μ \to b + v \to c .$ Then which of the following is true

μ, \frac{λ}{2}, v

are in A.P.

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λ, μ, v

are in A.P.

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λ, μ, v

are in H.P.

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λ, μ, v

are in G.P.

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Solution

The correct option is A $μ, \frac{λ}{2}, v$ are in A.P.
We have,
$\to r = 3^i + 2^j - 5^k \dots (i)$
and $\to r = λ \to a + μ \to b + v \to c$
$\Rightarrow \to r = λ (2^i -^j +^k) + μ (^i + 3^j - 2^k) + v (- 2^i +^j - 3^k)$
$\Rightarrow \to r =^i (2 λ + μ - 2 v) +^j (- λ + 3 μ + v) +^k (λ - 2 μ - 3 v) \dots (i i)$
Comparing like vectors, we get
$2 λ + μ - 2 v = 3 \dots (i i i)$
$- λ + 3 μ + v = 2 \dots (i v)$
$λ - 2 μ - 3 v = - 5 \dots (v)$
Solving these above equations, we get
$μ = 1, v = 2, λ = 3$
Hence, $μ, \frac{λ}{2}, v$ are in A.P.

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If →r=3^i+2^j−5^k, →a=2^i−^j+^k, →b=^i+3^j−2^k and →c=−2^i+^j−3^k such that →r=λ→a+μ→b+v→c. Then which of the following is true

If $\to r = 3^i + 2^j - 5^k, \to a = 2^i -^j +^k, \to b =^i + 3^j - 2^k$ and $\to c = - 2^i +^j - 3^k$ such that $\to r = λ \to a + μ \to b + v \to c .$ Then which of the following is true