The vector equation of the plane passing thro | vedclass

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Question

(A) Let the position vectors of the given points be $\vec{a} = \hat{i} - 2\hat{j} + 5\hat{k}$,$\vec{b} = -5\hat{j} - \hat{k}$,and $\vec{c} = -3\hat{i}

Vedclass · Accepted Answer

$\bar{r}=(1-\lambda-4 \mu) \bar{i}-(2+3 \lambda-7 \mu) \bar{j}+(5-6 \lambda-5 \mu) \bar{k}$

Vedclass · Answer

(A) Let the position vectors of the given points be $\vec{a} = \hat{i} - 2\hat{j} + 5\hat{k}$,$\vec{b} = -5\hat{j} - \hat{k}$,and $\vec{c} = -3\hat{i} + 5\hat{j}$.The vector equation of a plane passing through three points $\vec{a}$,$\vec{b}$,and $\vec{c}$ is given by $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a}) + \mu(\vec{c} - \vec{a})$.First,calculate the direction vectors:$\vec{b} - \vec{a} = (-5\hat{j} - \hat{k}) - (\hat{i} - 2\hat{j} + 5\hat{k}) = -\hat{i} - 3\hat{j} - 6\hat{k}$.$\vec{c} - \vec{a} = (-3\hat{i} + 5\hat{j}) - (\hat{i} - 2\hat{j} + 5\hat{k}) = -4\hat{i} + 7\hat{j} - 5\hat{k}$.Substituting these into the equation:$\vec{r} = (\hat{i} - 2\hat{j} + 5\hat{k}) + \lambda(-\hat{i} - 3\hat{j} - 6\hat{k}) + \mu(-4\hat{i} + 7\hat{j} - 5\hat{k})$.Grouping the components:$\vec{r} = (1 - \lambda - 4\mu)\hat{i} + (-2 - 3\lambda + 7\mu)\hat{j} + (5 - 6\lambda - 5\mu)\hat{k}$.This can be written as $\vec{r} = (1 - \lambda - 4\mu)\hat{i} - (2 + 3\lambda - 7\mu)\hat{j} + (5 - 6\lambda - 5\mu)\hat{k}$.

The vector equation of the plane passing through the points $(1, -2, 5)$,$(0, -5, -1)$ and $(-3, 5, 0)$ is

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