मान लीजिए overrightarrow{u}, overrightarrow{v | vedclass

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(C) दिया गया है,$|\overrightarrow{u}|=1, |\overrightarrow{v}|=1, \overrightarrow{u} \cdot \overrightarrow{v} 
eq 0, \overrightarrow{u} \cdot \overrig

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(C) दिया गया है,$|\overrightarrow{u}|=1, |\overrightarrow{v}|=1, \overrightarrow{u} \cdot \overrightarrow{v} 
eq 0, \overrightarrow{u} \cdot \overrightarrow{w}=1, \overrightarrow{v} \cdot \overrightarrow{w}=1, \overrightarrow{w} \cdot \overrightarrow{w}=4$ है।समांतर षट्फलक का आयतन अदिश त्रिक गुणनफल $|[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}]| = \sqrt{2}$ द्वारा दिया जाता है।अतः,$[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}]^2 = 2$ है।ग्राम सारणिक (Gram determinant) के गुण का उपयोग करते हुए:$[\overrightarrow{u} \overrightarrow{v} \overrightarrow{w}]^2 = \begin{vmatrix} \overrightarrow{u} \cdot \overrightarrow{u} & \overrightarrow{u} \cdot \overrightarrow{v} & \overrightarrow{u} \cdot \overrightarrow{w} \ \overrightarrow{v} \cdot \overrightarrow{u} & \overrightarrow{v} \cdot \overrightarrow{v} & \overrightarrow{v} \cdot \overrightarrow{w} \ \overrightarrow{w} \cdot \overrightarrow{u} & \overrightarrow{w} \cdot \overrightarrow{v} & \overrightarrow{w} \cdot \overrightarrow{w} \end{vmatrix} = \begin{vmatrix} 1 & \overrightarrow{u} \cdot \overrightarrow{v} & 1 \ \overrightarrow{u} \cdot \overrightarrow{v} & 1 & 1 \ 1 & 1 & 4 \end{vmatrix} = 2$ है।सारणिक का विस्तार करने पर:$1(4 - 1) - (\overrightarrow{u} \cdot \overrightarrow{v})(4\overrightarrow{u} \cdot \overrightarrow{v} - 1) + 1(\overrightarrow{u} \cdot \overrightarrow{v} - 1) = 2$ है।$3 - 4(\overrightarrow{u} \cdot \overrightarrow{v})^2 + \overrightarrow{u} \cdot \overrightarrow{v} + \overrightarrow{u} \cdot \overrightarrow{v} - 1 = 2$ है।$-4(\overrightarrow{u} \cdot \overrightarrow{v})^2 + 2(\overrightarrow{u} \cdot \overrightarrow{v}) = 0$ है।$2(\overrightarrow{u} \cdot \overrightarrow{v})(1 - 2(\overrightarrow{u} \cdot \overrightarrow{v})) = 0$ है।चूँकि $\overrightarrow{u} \cdot \overrightarrow{v} 
eq 0$,इसलिए $\overrightarrow{u} \cdot \overrightarrow{v} = \frac{1}{2}$ है।अब,$|3\overrightarrow{u} + 5\overrightarrow{v}|^2 = 9|\overrightarrow{u}|^2 + 25|\overrightarrow{v}|^2 + 30(\overrightarrow{u} \cdot \overrightarrow{v}) = 9(1) + 25(1) + 30(\frac{1}{2}) = 9 + 25 + 15 = 49$ है।अतः,$|3\overrightarrow{u} + 5\overrightarrow{v}| = \sqrt{49} = 7$ है।

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