Question 26a9b

Jun 26, 2017

Here positions of three points in 3D are given as (taking unit of length as $m$)

$P \to \left(4 , - 1 , 5\right)$

${P}_{1} \to \left(- 1 , - 1 , 0\right)$

${P}_{2} \to \left(3 , - 1 , - 3\right)$

So
$\vec{P {P}_{1}} = - 5 \hat{i} - 5 \hat{k}$

$\vec{P {P}_{2}} = - \hat{i} - 8 \hat{k}$

$\vec{{P}_{1} {P}_{2}} = 4 \hat{i} - 3 \hat{k}$

If the positon of bird ,when it is on line ${P}_{1} {P}_{2}$ be $O$.

1st method

Let $O {P}_{1} : O {P}_{2} = m : n$

So $\vec{P O} = \frac{n \cdot \vec{P {P}_{1}} + m \cdot \vec{P {P}_{2}}}{m + n}$

$\implies \vec{P O} = \frac{n \left(- 5 \hat{i} - 5 \hat{k}\right) + m \left(- \hat{i} - 8 \hat{k}\right)}{m + n}$

$\implies \vec{P O} = \frac{\left(- 5 n - m\right) \hat{i} + \left(- 5 n - 8 m\right) \hat{k}}{m + n}$

Now $\vec{P O} \mathmr{and} \vec{{P}_{1} {P}_{2}}$ are perpendicularly inclined.

So $\vec{P O} \cdot \vec{{P}_{1} {P}_{2}} = 0$

$\implies \frac{4 \left(- 5 n - m\right) - 3 \left(- 5 n - 8 m\right)}{m + n} = 0$

$\implies - 20 n - 4 m + 15 n + 24 m = 0$

$\implies 20 m = 5 n$

$\implies \frac{m}{n} = \frac{1}{4}$

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2nd method

Then $\vec{P O}$ will be perpendicular to $\vec{{P}_{1} {P}_{2}}$

So $\vec{O {P}_{1}}$ is the projection of $\vec{P {P}_{1}}$ on vec (P_2P_1

Hence $\left\mid \vec{O {P}_{1}} \right\mid = \vec{{P}_{2} {P}_{1}} \cdot \vec{P {P}_{1}}$
$= \frac{\left(- 4 \hat{i} + 3 \hat{k}\right) \cdot \left(- 5 \hat{i} - 5 \hat{k}\right)}{\left\mid - 4 \hat{i} + 3 k \right\mid}$

$= \frac{20 - 15}{\sqrt{{\left(- 4\right)}^{2} + {3}^{2}}} = 1$

Again $\vec{O {P}_{2}}$ is the projection of $\vec{P {P}_{2}}$ on vec (P_1P_2

$\left\mid \vec{O {P}_{2}} \right\mid = \vec{{P}_{1} {P}_{2}} \cdot \vec{P {P}_{1}}$
$= \frac{\left(4 \hat{i} - 3 \hat{k}\right) \cdot \left(- \hat{i} - 8 \hat{k}\right)}{\left\mid 4 \hat{i} - 3 \hat{k} \right\mid}$

$= \frac{- 4 + 24}{\sqrt{{4}^{2} + {\left(- 3\right)}^{2}}} = 4$

So $\vec{P O}$ divides ${P}_{1} {P}_{2}$ ar $O$ in the ratio $O {P}_{1} : O {P}_{2} = 1 : 4$

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Finally

$\vec{P O} = \frac{4 \cdot \vec{P {P}_{1}} + 1 \cdot \vec{P {P}_{2}}}{1 + 4}$

$\implies \vec{P O} = \frac{4 \left(- 5 \hat{i} - 5 \hat{k}\right) + 1 \left(- \hat{i} - 8 \hat{k}\right)}{5}$

$\implies \vec{P O} = \frac{- 20 \hat{i} - 20 \hat{k} - \hat{i} - 8 \hat{k}}{5}$

$\implies \vec{P O} = \frac{- 21 \hat{i} - 28 \hat{k}}{5}$

$\implies \left\mid \vec{P O} \right\mid = \frac{1}{5} \left\mid - 21 \hat{i} - 28 \hat{k} \right\mid$

$\implies \left\mid \vec{P O} \right\mid = \frac{1}{5} \cdot \sqrt{{\left(- 21\right)}^{2} + {\left(- 28\right)}^{2}}$

$\implies \left\mid \vec{P O} \right\mid = \frac{1}{5} \cdot \sqrt{1225} = 7$ m
Time (t) taken by the bird to cover the distance $P O$ is

t="distance PO"/"velocity of the bird"=(7m)/(2m"/"s)=3.5# s