Question

If three vectors ${\displaystyle \overline{a},\overline{b},\overline{c}}$ are not coplanar and ${\displaystyle x\overline{a}+y\overline{b}+z\overline{c}=\overline{0}}$, then prove that x = y = z = 0

Answer 1

Suppose x ≠ 0

${\displaystyle x\overline{a}+y\overline{b}+z\overline{c}=\overline{0}}$     ...(Given)

∴ ${\displaystyle x\overline{a}=-y\overline{b}-z\overline{c}}$

∴  ${\displaystyle \overline{a}=(-\frac{y}{x})\overline{b}-\left(\frac{z}{x}\right)\overline{c}}$    ...[∵ x ≠ 0]

∴ ${\displaystyle \overline{a}}$ is a linear combination of ${\displaystyle \overline{b}}$ and ${\displaystyle \overline{c}}$

∴  ${\displaystyle \overline{a},\overline{b},\overline{c}}$ are coplanar

This is a contradiction

∴ x = 0

Similarly, y = z = 0

∴ x = y = z = 0