Vector and Cartesian Equation of a Plane

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

2x + 3y + 4z – 12 = 0

Find the vector equation of the plane with intercepts 3, –4 and 2 on x, y and z-axis respectively.

Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes \[\vec{r} \cdot \left( \hat{i}  - \hat{j} + 2 \hat{k}  \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j}  + 2 \hat{k} \right) = 6\]

Find the vector equation of the plane passing through three points with position vectors ` hati+hatj-2hatk , 2hati-hatj+hatk and hati+2hatj+hatk` . Also find the coordinates of the point of intersection of this plane and the line `vecr=3hati-hatj-hatk lambda +(2hati-2hatj+hatk)`

Find the vector and Cartesian forms of the equation of the plane passing through the point (1, 2, −4) and parallel to the lines \[\vec{r} = \left( \hat{i} + 2 \hat{j}  - 4 \hat{k}  \right) + \lambda\left( 2 \hat{i}  + 3 \hat{j}  + 6 \hat{k}  \right)\] and \[\vec{r} = \left( \hat{i}  - 3 \hat{j}  + 5 \hat{k}  \right) + \mu\left( \hat{i}  + \hat{j}  - \hat{k} \right)\] Also, find the distance of the point (9, −8, −10) from the plane thus obtained.  

Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.

The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector  `2hati + hatj + 2hatk.`