Show that vectors 2-k-3-5k- and 3-4-4k- form the vertices of the triangle-

We have, #10; #10;Given three vectors #10; #10;Let, #10; #10; OA=2i k #10; #10; OB=i 3j 5k #10; #10; #10; #10;OC=3i 4j 4k ...

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Standard XII

Mathematics

Section Formula

Show that vec...

Question

Show that vectors $2^i -^k,^i - 3^j - 5^k$ and $3^i - 4^j - 4^k$ form the vertices of the triangle.

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Solution

We have,

Given three vectors

Let,

$- - \to O A = 2 ˆ i - ˆ k$

$- - \to O B = ˆ i - 3 ˆ j - 5 ˆ k$

$- - \to O C = 3 ˆ i - 4 ˆ j - 4 ˆ k$

So,

$- - \to A B = - - \to O B - - - \to O A$

$- - \to A B = (ˆ i - 3 ˆ j - 5 ˆ k) - (2 ˆ i - ˆ k)$

$- - \to A B = - ˆ i - 3 ˆ j - 4 ˆ k$

$- - \to B C = - - \to O C - - - \to O B$

$- - \to B C = (3 ˆ i - 4 ˆ j - 4 ˆ k) - (ˆ i - 3 ˆ j - 5 ˆ k)$

$- - \to B C = 2 ˆ i - ˆ j + ˆ k$

$- - \to C A = - - \to O A - - - \to O C$

$- - \to C A = (2 ˆ i - ˆ k) - (3 ˆ i - 4 ˆ j - 4 ˆ k)$

$- - \to C A = - ˆ i + 4 ˆ j + 3 ˆ k$

Now,

$∣ ∣ ∣ - - \to A B ∣ ∣ ∣ = \sqrt{{(- 1)}^{2} + {(- 3)}^{2} + {(- 4)}^{2}}$

$∣ ∣ ∣ - - \to A B ∣ ∣ ∣ = \sqrt{26}$

$∣ ∣ ∣ - - \to B C ∣ ∣ ∣ = \sqrt{2^{2} + {(- 1)}^{2} + 1^{2}} = \sqrt{6}$

$\begin{align}

$∣ ∣ ∣ - - \to C A ∣ ∣ ∣ = \sqrt{{(- 1)}^{2} + 4^{2} + 3^{2}}$

$∣ ∣ ∣ - - \to C A ∣ ∣ ∣ = \sqrt{1 + 16 + 9} = \sqrt{26}$

Now,

$A B = C A$

Hence, it is a isosceles triangle.

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Show that vectors 2^i−^k,^i−3^j−5^k and 3^i−4^j−4^k form the vertices of the triangle.

Show that vectors $2^i -^k,^i - 3^j - 5^k$ and $3^i - 4^j - 4^k$ form the vertices of the triangle.