Question

$A(1,2,3)$ , $B(1,2,5)$ , $C(1,4,3)$ are vertices of $\Delta ABC$ then position vector of orthocentre is

• A. $(2,3,4)$
• B. $(1,2,3)$
• C. $(1,5,4)$
• D. $(1,3,4)$

Answer 1

Answer: (B)

$(1,2,3)$

$\overrightarrow{AB}=\overrightarrow{B}-\overrightarrow{A}=2\overrightarrow{k}$
$\overrightarrow{AC}=\overrightarrow{C}-\overrightarrow{A}=2\overrightarrow{j}$
$\overrightarrow{BC}=\overrightarrow{C}-\overrightarrow{B}=2\overrightarrow{j}-2\overrightarrow{k}$
Now
$\overrightarrow{AB}\cdot \overrightarrow{AC}=2\overrightarrow{k}\cdot 2\overrightarrow{j}$
$=0$
Thus, $AB$ is perpendicular to $AC$
$\therefore △ABC$ is a right angled triangle and right angled at A.
$\therefore A$ is the orthocentre.

Hence, $(1,2,3)$ is the orthocentre.