Show that the points A- B- C with position vectors 2- - - - k- - - 3- - 5k- and 3- - 4- - 4k- respectively- are the vertices of a right-angled triangle- Hence- find the area of the triangle-

Vertices of Δ ABC with position vectors, #160; #160; #160; #160; [ OA =2 i j + k #160;; OB = i 3 j 5 k ...

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Byju's Answer

Standard XII

Physics

Position Vector

Show that the...

Question

Show that the points $A, B, C$ with position vectors $2^i -^j +^k,^i - 3^j - 5^k$ and $3^i - 4^j - 4^k$ respectively, are the vertices of a right-angled triangle. Hence, find the area of the triangle.

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Solution

Vertices of $Δ A B C$ with position vectors,
$\begin{matrix} ¯ ¯¯¯¯¯¯ ¯ O A = 2 ˆ i - ˆ j + ˆ k ¯ ¯¯¯¯¯¯ ¯ O B = ˆ i - 3 ˆ j - 5 ˆ k ¯ ¯¯¯¯¯¯ ¯ O C = 3 ˆ i - 4 ˆ j - 4 ˆ k \end{matrix}$
the $Δ A B C$ is right angled triangle and find the area of $Δ A B C$
then,
$\begin{matrix} - - \to A B = - - \to O B - - - \to O A = ˆ (i - 3 ˆ j - 5 ˆ k) - (2 ˆ i - ˆ j + ˆ k) = ˆ i - 3 ˆ j - 5 ˆ k - 2 ˆ i + ˆ j - ˆ k = - ˆ i - 2 ˆ j + 6 ˆ k, ∴ ∣ ∣ ¯ ¯¯¯¯¯¯ ¯ A B ∣ ∣ = \sqrt{41} - - \to B C = - - \to O C - - - \to O B = (3 ˆ i - 4 ˆ j - 4 ˆ k) - (ˆ i - 3 ˆ j - 5 ˆ k) = 3 ˆ i - 4 ˆ j - 4 ˆ k - ˆ i + 3 ˆ j + 5 ˆ k = 2 ˆ i - ˆ j + ˆ k ∴ ∣ ∣ ∣ - - \to B C ∣ ∣ ∣ = \sqrt{6} - - \to C A = - - \to O A - - - \to O C = (2 ˆ i - ˆ j + ˆ k) - (3 ˆ i - 4 ˆ j - 4 ˆ k) = 2 ˆ i - ˆ j + ˆ k - 3 ˆ i + 4 ˆ j + 4 ˆ k = - ˆ i + 3 ˆ j + 5 ˆ k ∴ ∣ ∣ ∣ - - \to C A ∣ ∣ ∣ = \sqrt{35} \end{matrix}$

$\begin{matrix} b y P a t h a g o r a s t h e o r a m, a^{2} = b^{2} + c^{2} s o, {∣ ∣ ¯ ¯¯¯¯¯¯ ¯ A B ∣ ∣}^{2} = {∣ ∣ ∣ - - \to B C ∣ ∣ ∣}^{2} + {∣ ∣ ∣ - - \to C A ∣ ∣ ∣}^{2} = 6 + 35 = 41 ∣ ∣ ¯ ¯¯¯¯¯¯ ¯ A B ∣ ∣ = \sqrt{41} \end{matrix}$
So,
$Δ A B C$ is right angled triangle at C.
and the Area of $Δ A B C$ = $\begin{matrix} \frac{1}{2} ∣ ∣ ∣ - - \to B C ∣ ∣ ∣ \times ∣ ∣ ∣ - - \to C A ∣ ∣ ∣ = \frac{1}{2} \sqrt{6} \times \sqrt{35} = \frac{\sqrt{210}}{2} \end{matrix}$

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Similar questions

Show that the points A,B,C with position vectors $2^i -^j +^k,^i - 3^j - 5^k,$ and $3^i - 4^j - 4^k$ respectively,are the vertices of a right-angled triangle. Hence find the area of the rectangle.

Q. Show that the points

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with position vectors

2^i -^j +^k,^i - 3^j - 5^k

and

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respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle.

Show that the points A,B and C with position vectors, $a = 3^i - 4^j - 4^k, b = 2^i -^j +^k a n d c =^i - 3^j - 5^k$ respectively, form the vertices of a right angled triangle.