Q.A plane makes non zero intercepts a, b, c on the co-ordinates axes. Show that the vector equation of the plane is $\vec{r}\cdot(bc\hat{i}+ca\hat{j}+ab\hat{k})=abc$.

[!TLDR] Intercept form $\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$; clear denominators by multiplying by $abc$. …

A plane making non-zero intercepts $a,b,c$ on the coordinate axes meets them at $A(a,0,0)$, $B(0,b,0)$, $C(0,0,c)$, and has the well-known intercept form

Multiplying both sides by $abc$ clears the denominators:

In vector form, writing $\vec r=x\hat i+y\hat j+z\hat k$, this is exactly

$\vec r\cdot(bc\hat i+ca\hat j+ab\hat k)=abc,$ …

Start from the standard Cartesian intercept form of a plane, clear the denominators by multiplying through by $abc$, then recognise the resulting l …

Misassigning which coefficient goes with which axis while clearing denominators (the coefficient of $x$ is $bc$, not $ab$ o …