if a = 2 i + j - k , b = 4 i - 7 j + k , find a vector c such tha… - Vidyadip

Question

$\text{if } \vec{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}, \vec{\text{b}} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}, \text{find a vector } \vec{\text{c}} \text{ such that } \vec{\text{a}} \times \vec{\text{c}}=\vec{\text{b }} \text{ and } \vec{\text{a }} . \vec{\text{c}} = 6.$

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Answer

$\text{Let } \vec{\text{c}} = \text{x}\hat{\text{i}} + \text{y}\hat{\text{j}} + \text{z} \hat{\text{k }} ; \vec{\text{ a}}.\vec{\text{c}} = 6 \Rightarrow \text{2x + y - z = 6} $ $\text{Now}, \vec{\text{a}} \times \vec{\text{c}} = \vec{\text{b}} \Rightarrow\begin{vmatrix} \hat{\text{i}} & \hat{\text{j}} & \hat{\text{k}} \\ 2 & 1 & -1 \\ \text{x} & \text{y} & \text{z} \end{vmatrix} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}$$\Rightarrow \hat{\text{i}} \text{(z + y)} - \hat{\text{j}} \text{(2z + x)} + \hat{\text{k}} \text{(2y - x)} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}$
$\Rightarrow \text{z + y = 4, 2z + x = 7, 2y - x = 1}$ Solving and getting $\text{ = 3, y = 2, z = 2}$ $\vec{\text{c}} = 3\hat{\text{i}} + 2\hat{\text{i}} + 2\hat{\text{k}}$

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