For non-coplanar vectors a, b and c,if the po | vedclass

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(C) The equation of the plane is given by $r = b + c + x(a - b) + y(c + a) = (x + y)a + (1 - x)b + (1 + y)c$ $\ldots(i)$.The equation of the line is g

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$2$

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(C) The equation of the plane is given by $r = b + c + x(a - b) + y(c + a) = (x + y)a + (1 - x)b + (1 + y)c$ $\ldots(i)$.The equation of the line is given by $r = a + t(b - c) = a + tb - tc$ $\ldots(ii)$.Since the point of intersection satisfies both equations,we equate the coefficients of $a, b,$ and $c$ because $a, b, c$ are non-coplanar:$x + y = 1$ $\ldots(iii)$$1 - x = t$ $\ldots(iv)$$1 + y = -t$ $\ldots(v)$Adding equations $(iv)$ and $(v)$,we get $2 - x + y = 0$,so $x - y = 2$ $\ldots(vi)$.Adding $(iii)$ and $(vi)$,we get $2x = 3$,so $x = \frac{3}{2}$.Substituting $x = \frac{3}{2}$ into $(iii)$,we get $y = 1 - \frac{3}{2} = -\frac{1}{2}$.Substituting $x = \frac{3}{2}$ into $(iv)$,we get $t = 1 - \frac{3}{2} = -\frac{1}{2}$.Now,the point of intersection $r$ is $a + t(b - c) = a - \frac{1}{2}(b - c) = a - \frac{1}{2}b + \frac{1}{2}c$.Comparing this with $la + mb + nc$,we get $l = 1, m = -\frac{1}{2}, n = \frac{1}{2}$.Finally,$3l + 4m + 2n = 3(1) + 4(-\frac{1}{2}) + 2(\frac{1}{2}) = 3 - 2 + 1 = 2$.

For non-coplanar vectors $a, b$ and $c$,if the point of intersection of the line $r=a+t(b-c)$ and the plane $r=b+c+x(a-b)+y(c+a)$ is $l a+m b+n c$,then $3 l+4 m+2 n=$

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