Given-p-2-i-2-3-j-3-k-amp-q-a-i-j-2k-If-proj-q-p-2-2-9-q-then-q-

Question Number 144166 by liberty last updated on 22/Jun/21

Given \vec{p} = \sqrt{2} \hat{i} + 2 \sqrt{3} \hat{j} + \sqrt{3} \hat{k} &

\vec{q} = a \hat{i} + \hat{j} + 2 \hat{k} . If {proj}_{\vec{q}} \vec{p} = \frac{2 \sqrt{2}}{9} \vec{q}

then ∣ \vec{q} ∣ = ?

Answered by benjo_mathlover last updated on 23/Jun/21

[\frac{\vec{p} . \vec{q}}{∣ \vec{q} ∣^{2}}] \vec{q} = \frac{2 \sqrt{2}}{9} \vec{q}

[\frac{a \sqrt{2} + 2 \sqrt{3} + 2 \sqrt{3}}{5 + a^{2}}] = \frac{2 \sqrt{2}}{9}

9 a \sqrt{2} + 36 \sqrt{3} = 10 \sqrt{2} + 2 \sqrt{2} a^{2}

2 \sqrt{2} a^{2} - 9 \sqrt{2} a + 10 \sqrt{2} - 36 \sqrt{3} = 0

a = \frac{9 \sqrt{2} \pm \sqrt{162 - 4 (40 - 73 \sqrt{6})}}{4 \sqrt{2}}

a = \frac{9 \sqrt{2} \pm \sqrt{292 \sqrt{6} + 2}}{4 \sqrt{2}}