Given- three non - zero- non -coplanar vectors a- b- c- and r-1-pa-qb-c- and r-2-a-pb-qc- if the vectors r-1-2r-2- and 2r-1-r-2- are collinear then p-q is

The correct option is C (1,1) r_1+2r_2 =(p+2)a+(q+2p)b+(1+2q)c Similarly 2r_1+2r_2 =(2p+1)a+(2q+p)b+(2+q)c Since they are collinear. ...

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Test for Collinearity of Vectors

Given, three ...

Question

Given, three non - zero, non -coplanar vectors $\to a$ , $\to b$ , $\to c$ and $_{1} = p \to a + q \to b + \to c$ and $_{2} = \to a + p \to b + q \to c$ , if the vectors $_{1} + 2_{2}$ and $2_{1} +_{2}$ are collinear then $(p, q)$ is

(0, 0)

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(1, - 1)

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(- 1, 1)

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(1, 1)

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Solution

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\to a, \to b

\to c

_{1} = p \to a + q \to b + \to c

_{2} = \to a + p \to b + q \to c

_{1} + 2_{2}

2_{1} +_{2}

(p, q)

\to a, \to b, \to c

_{1} = \to a - \to b + \to c,_{2} = \to b + \to c - \to a

_{3} = \to c + \to a - \to b, \to r = 2 \to a - 2 \to b + 4 \to c

\to r = x_{1}_{1} + x_{2}_{2} + x_{3}_{3}

\to a, \to b, \to c

_{1} = \to a - \to b + \to c,_{2} = \to b + \to c - \to a

_{3} = \to c + \to a + \to b, \to r = 2 \to a - 3 \to b + 4 \to c

\to r = λ_{1}_{1} + λ_{2}_{2} + λ_{3}_{3}

\to a, \to b, \to c

_{1} = \to a - \to b + \to c,_{2} = \to b + \to c - \to a,_{3} = \to c + \to a + \to b

\to r = 2 \to a - 3 \to b + 4 \to c

\to r = λ_{1}_{1} + λ_{2}_{2} + λ_{3}_{3}

\to a, \to b

\to c

\frac{π}{3} .

\to a \times \to b + \to b \times \to c = p \to a + q \to b + r \to c,

p, q

r

\frac{p^{2} + 2 q^{2} + r^{2}}{q^{2}}

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