Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The equation of the plane is given by $\vec{r}\cdot \vec{n}=d$.

For the intercept on the x-axis, the position vector of the point is $\vec{r}=x\hat{i}$.

Substituting this into the equation of the plane, we get:

$(x\hat{i})\cdot \vec{n}=d$

$x(\hat{i}\cdot \vec{n})=d$

$x=\frac{d}{\hat{i}\cdot \vec{n}}$

Similarly, for the intercept on the y-axis, the position vector is $\vec{r}=y\hat{j}$. Substituting this gives:

$(y\hat{j})\cdot \vec{n}=d\Rightarrow y=\frac{d}{\hat{j}\cdot \vec{n}}$

For the intercept on the z-axis, the position vector is $\vec{r}=z\hat{k}$. Substituting this gives:

$(z\hat{k})\cdot \vec{n}=d\Rightarrow z=\frac{d}{\hat{k}\cdot \vec{n}}$

Thus, the intercepts on the coordinate axes are $\frac{d}{\hat{i}\cdot \vec{n}},\frac{d}{\hat{j}\cdot \vec{n}},\frac{d}{\hat{k}\cdot \vec{n}}$.

Answer: $\frac{d}{\hat{i}\cdot \vec{n}},\frac{d}{\hat{j}\cdot \vec{n}},\frac{d}{\hat{k}\cdot \vec{n}}$