If → a = ^ i + ^ j + ^ k , → b = 2 ^ i − ^ j + ^ k and → c = ^ i + x ^ j + y ^ k , are linearly dependent and | → c | = √ 3 then ( x , y ) is Options: A . &nbsp(1,1) B . &nbsp(−2,0) C . &nbsp(15,75) D . &nbsp(−75,35)

Answer: Option A : A Given that the three vectors are linearly dependent so → c = l → a + m → b ⇒ l + 2 m = 1 l − m = x ⇒ x = 3 y − 2 l + m = y Also, x 2 + y 2 + 1 = 3 10 y 2 − 12 y + 2 = 0 ⇒ y = 1 , 1 5 x = 1 , − 7 5

If →a=^i+^j+^k,→b=2^i−^j+^k and →c=^i+x^j+y^k, are l

Question

\to a =^i +^j +^k, \to b = 2^i -^j +^k

and

\to c =^i + x^j + y^k

, are linearly dependent and

| \to c | = \sqrt{3}

then

(x, y)

Options:

A . (1,1)

B . (−2,0)

C . (15,75)

D . (−75,35)

Answer: Option A
:
A
Given that the three vectors are linearly dependent so

\to c = l \to a + m \to b

\Rightarrow l + 2 m = 1

l - m = x

\Rightarrow x = 3 y - 2

l + m = y

Also,

x^{2} + y^{2} + 1 = 3

10 y^{2} - 12 y + 2 = 0

\Rightarrow y = 1, \frac{1}{5}

x = 1, - \frac{7}{5}

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