First slide

Geometry in vectors

Question

If $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are non - zero coplanar vectors and $2 x_{1} \vec{a} + 2 x_{2} \vec{b} - \vec{c} + 3 x_{3} \vec{d} = 0$ then

Moderate

A

$x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \geq \frac{1}{2}$
B

$x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \geq \frac{1}{17}$
C

$x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \geq \frac{1}{20}$
D

$x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \geq \frac{1}{34}$

Solution

$\begin{array}{l} 2 x_{1} + 2 x_{2} + 3 x_{3} = 1 \\ ∴ |2 x_{1} + 2 x_{2} + 3 x_{3}| \leq \sqrt{4 + 4 + 9} \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \\ \Rightarrow x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \geq \frac{1}{17} \end{array}$

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