The equation of the plane passing through the point $\left(1,2,3\right)$ and having the normal vector $\hat{N}=3\hat{i}-\hat{j}+2\hat{k}$ is

Line and Plane — Preparation

This question tests your understanding of Line and Plane, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

This topic extends coordinate geometry into three dimensions. It covers direction cosines and direction ratios of a line, the equation of a line in vector and Cartesian form, the angle between two lines, the shortest distance between two skew lines, and the condition for two lines to intersect. It also covers the equation of a plane in its different forms (normal form, through a point perpendicular to a vector, through three points, and intercept form), the angle between two planes and between a line and a plane, the distance of a point from a plane, and the distance between parallel planes. Remember that the direction cosines l, m and n of any line satisfy l^2 + m^2 + n^2 = 1, and that the distance of a point (x1, y1, z1) from the plane ax + by + cz + d = 0 is |ax1 + by1 + cz1 + d| / sqrt(a^2 + b^2 + c^2). Practise the line and plane questions below, each with a step-by-step solution.