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Mathematics Question on Three Dimensional Geometry

Find the length of perpendicular from origin to the plane r⃗⋅(3i^−4j^−12k^)+39=0\vec{r}\cdot\left(3\hat{i}-4\hat{j}-12\hat{k}\right)+39=0

A

11

B

33

C

17\frac{1}{7}

D

none of these

Answer

33

Explanation

Solution

The equation of the plane is r⃗⋅(3i^−4j^−12k^)=−39\vec{r}\cdot\left(3\hat{i}-4\hat{j}-12\hat{k}\right)=-39 or r⃗⋅(−3i^+4j^+12k^)=39\vec{r}\cdot\left(-3\hat{i}+4\hat{j}+12\hat{k}\right)=39 Now, ∣n⃗∣=(−3)2+42+122=13\left|\vec{n}\right|=\sqrt{\left(-3\right)^{2}+4^{2}+12^{2}}=13 ∴\therefore Required distance d=39∣n⃗∣=3913=3d=\frac{39}{\left|\vec{n}\right|}=\frac{39}{13}=3. Hence, the length of perpendicular from origin to the given plane is 33.