State whether the given statement is true or false If x- 0 be any real number - r1- x- 2x- -3x r2- 2x-1- 2x-3- x-1 and r3- 3x-5- x-5- x-2If the vectors r1- r2- r3 are coplanar-

( r_1 r_2 r_3 )= | #160;x amp; #160;2x amp; 3x #160; #160;2x+1 amp; #160;2x+3 amp; x+1 #160; #160;3x+5 amp; #160 ...

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Byju's Answer

Standard XII

Mathematics

Condition for Coplanarity of Four Points

State whether...

Question

State whether the given statement is true or false If $x \neq 0$ be any real number . $r_{1} = x, 2 x, - 3 x$ $r_{2} = 2 x + 1, 2 x + 3, x + 1$ and $r_{3} = 3 x + 5, x + 5, x + 2$
If the vectors $r_{1}, r_{2}, r_{3}$ are coplanar.

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Solution

$(r_{1} r_{2} r_{3}) = ∣ ∣ ∣ ∣ \begin{matrix} x & 2 x & - 3 x 2 x + 1 & 2 x + 3 & x + 1 3 x + 5 & x + 5 & x + 2 \end{matrix} ∣ ∣ ∣ ∣$
Expand the above determinant
$∴ (r_{1} r_{2} r_{3}) = x (15 x^{2} + 31 x + 37)$
Now quadratic expression $15 x^{2} + 31 x + 37$ is such that its
$Δ = {(31)}^{2} - 60 (37) = - i v e$
and hence its sign is same as that of the coefficient of first term i.e.,15 i.e, +ive.
Also $x \neq 0$ is a real number.
$∴ (r_{1} . r_{2} . r_{3}) \neq 0.$
Hence the vectors are non-coplanar.

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