A, B and C are the points (1, 1, 1), (3, 2, 3) and (2, 2, 1) respectively. The length of the perpendicular from C to the side AB of the triangle ABC is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 1
The area of the triangle can be found by the vector product, and also as \(\displaystyle\frac{1}{2}\times\text{base}\times\text{height}\) with base AB.
\(\overrightarrow{AB} = 2\hat{\imath}+\hat{\jmath}+2\hat{k}, \quad |\overrightarrow{AB}| = \sqrt{4+1+4} = 3\)
\(\overrightarrow{AC} = \hat{\imath}+\hat{\jmath}\)
\( \displaystyle \overrightarrow{AB}\times\overrightarrow{AC} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ 2 & 1 & 2 \\ 1 & 1 & 0 \end{vmatrix} \)
\( = \hat{\imath}\,(0-2)-\hat{\jmath}\,(0-2)+\hat{k}\,(2-1) = -2\hat{\imath}+2\hat{\jmath}+\hat{k}\)
\(|\overrightarrow{AB}\times\overrightarrow{AC}| = \sqrt{4+4+1} = 3\)
\( \displaystyle \text{Area} = \frac{1}{2}\times3 = \frac{3}{2} \)
Let h be the perpendicular from C to AB:
\( \displaystyle \frac{1}{2}\times3\times h = \frac{3}{2} \)
\(h = 1\)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{3}{2}\) — Incorrect. This is the area of the triangle, not the perpendicular.
Option (2) \(\displaystyle\frac{1}{2}\) — Incorrect. This is the area divided by AB; the factor 2 from \(\displaystyle\frac{1}{2}\times\text{base}\times\text{height}\) is missed.
Option (3) 3 — Incorrect. This is \(|\overrightarrow{AB}\times\overrightarrow{AC}|\), which still has to be divided by \(|\overrightarrow{AB}| = 3\).