Assertion (A): a, b, c, d are position vector | vedclass

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Question

(A) The vector equation of a plane passing through three non-collinear points with position vectors $a, b, c$ is given by $r = (1 - x - y)a + xb + yc$

Vedclass · Accepted Answer

Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A)$

Vedclass · Answer

(A) The vector equation of a plane passing through three non-collinear points with position vectors $a, b, c$ is given by $r = (1 - x - y)a + xb + yc$. If four points $a, b, c, d$ are coplanar, then $d$ must be expressible as a linear combination of $a, b, c$ such that the sum of the coefficients is $1$. Given the equation $2a - 3b + 7c - 6d = 0$, we can rewrite it as $6d = 2a - 3b + 7c$, which implies $d = \frac{2}{6}a - \frac{3}{6}b + \frac{7}{6}c = \frac{1}{3}a - \frac{1}{2}b + \frac{7}{6}c$. The sum of the coefficients is $\frac{1}{3} - \frac{1}{2} + \frac{7}{6} = \frac{2 - 3 + 7}{6} = \frac{6}{6} = 1$. Since the sum of the coefficients is $1$, the point $d$ lies in the plane formed by $a, b, c$. Thus, $(A)$ is true, $(R)$ is true, and $(R)$ is the correct explanation of $(A)$.

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