Question

$A,B,C,D$ are four points on a plane with position vectors $\text{a,b,c and d}$ respectively such that $(a-d).(b-c)=(b-d).(c-d)=0$. For $△ABC,D$ is the?

• A. Incentre
• B. orthocentre
• C. centroid
• D. None of these

Answer 1

The correct option is B

orthocentre

Explanation for the correct option:

Using the concept of dot product:

According to the question,

$\because (a-d).(b-c)=0$

$\Rightarrow$ $DA.CB=0$

$\therefore AD\perp BC$ $\left[\because \overrightarrow{a}.\overrightarrow{b}=0\Rightarrow \overrightarrow{a}\text{is perpendicular to}\overrightarrow{b}\right]$

Again, $\because (b-d).(c-a)=0$

$\Rightarrow$ $DB.AC=0$

$\therefore BD\perp CA$

Thus, $D$ is the intersection of the altitudes passing through $A$ and $B$.

$\therefore D$ is the orthocentre of $△ABC$.

Hence, option ‘B’ is correct.