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Find the Cartesian equation of the plane $\;(a)\; \overrightarrow r. (\hat i + \hat j - \hat k) = 2$

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$\begin{array}{1 1} x+y-z=2 \\ x+y+z=-2 \\ x+y+z=2 \\ x+y-z=-2 \end{array} $

1 Answer

  • For any arbitrary point $P(x,y,z)$ on the plane,the position vector $\overrightarrow r$ is given by $\overrightarrow r=x\hat i+y\hat j+z\hat k$
  • The cartesian equation of the plane is of the form $lx+my+nz=d$
Step 1:
The given equation of the plane is $\overrightarrow r.(\hat i+\hat j-\hat k)=2$-----(1)
But we know $\overrightarrow r=x\hat i+y\hat j+z\hat k$
Substituting for $\overrightarrow r$ in equation (1) we get
$(x\hat i+y\hat j+z\hat k)(\hat i+\hat j-\hat k)=2$
Step 2:
Apply the dot product rule we get
This equation is the required Cartesian equation of the plane.
answered May 30, 2013 by sreemathi.v

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