# Find the Cartesian equation of the following planes: (c) $r .[(s - 2t)\hat i + (3- t) \hat j +(2 s + t) \hat k] = 15$

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com

Toolbox:
• 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 $\overrightarrow r.[(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k]=15$-----(1)
We know $\overrightarrow r=x\hat i+y\hat j+z\hat k$
Substituting for $\overrightarrow r$ in equ(1) we get
$(x\hat i+y\hat j+z\hat k).[(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k]=15$
Step 2:
We know that $\hat i.\hat i=\hat j.\hat j=\hat k.\hat k=1$
$(s-2t)x+(3-t)y+(2s+t)z=15$
This equation is the required Cartesian equation of the plane.