First slide

Direction ratios and direction cosines

Question

The triangle whose vertices are $A (3, - 1, 2), B (λ, - 1, 1), C (1, 1, - 2)$ , is a right-angled triangle. Then the sum of all possible integral values of $λ$ is

Moderate

A

$3$
B

$- 3$
C

$- 2$
D

$2$

Solution

Direction ratios of   $\bar{A B}, \bar{B C}, \bar{C A}$ are respectively are $〈3 - λ, 0, 1〉, 〈λ - 1, - 2, 3〉, 〈- 2, 2, - 4〉$ .
In right angle triangle, right angle may occur at any $B$ vertex    $\bar{A B}, \bar{B C}$
If the right angle is at the vertex then the dot product of  is zero.
$\begin{matrix} (3 - λ) (λ - 1) + 3 = 0 \\ 3 λ - 3 - λ^{2} + λ + 3 = 0 \\ λ^{2} - 4 λ = 0 \\ λ (λ - 4) = 0 \end{matrix}$
It gives $λ = 0, 4$
If the right angle is at the vertex $C$ , then the dot product of    $\bar{A C}, \bar{B C}$ is zero
$\begin{matrix} - 2 (3 - λ) - 4 - 12 = 0 \\ 2 λ - 2 + 16 = 0 \\ 2 λ + 14 = 0 \\ λ = - 7 \end{matrix}$
It gives $λ = - 7$
If the right angle is at the vertex $A$ then the dot product of    $\bar{A C}, \bar{A B}$ is zero
        $\begin{matrix} - 2 (3 - λ) + 2 (0) - 4 (1) = 0 \\ - 6 + 2 λ - 4 = 0 \\ 2 λ = 10 \\ λ = 5 \end{matrix}$
It gives $λ = 5$
Therefore, sum of all the possible values of $λ$ is $2$

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