Show that the vectors 2-k-3-5k- and 3-4-4k- form the vertices of a right-angled

Let A = 2î ĵ+k̂,B = î 3ĵ 5k̂ and C = 3î 4ĵ 4k̂ AB = (î 3ĵ 5k̂) (2î ĵ+k̂) ⇒AB= î 2ĵ 6k̂ BC=(3î 4ĵ 4k̂) (î 3ĵ 5k̂) ⇒BC = 2î ĵ+k̂ CA = (2î ĵ+k̂) (3î 4ĵ 4k̂) ...

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Byju's Answer

Standard IX

Mathematics

Plotting a Point

Show that the...

Question

Show that the vectors $2^i -^j +^k,^i - 3^j - 5^k and 3^i - 4^j - 4^k form the vertices of a right-angled$

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Solution

$Let A = 2^i -^j +^k, B =^i - 3^j - 5^k and C = 3^i - 4^j - 4^k$

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

$\Rightarrow - - \to A B = -^i - 2^j - 6^k$

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

$\Rightarrow - - \to B C = 2^i -^j +^k$

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

$\Rightarrow - - \to C A = -^i + 3^j + 5^k$

$- - \to B C . - - \to C A = (2^i -^j +^k) . (-^j + 3^j + 5^k)$

$- - \to B C . - - \to C A = (2 \times - 1) + (- 1 \times 3) + (1 \times 5)$

$- - \to B C . - - \to C A = - 5 + 5 = 0$

$\Rightarrow - - \to B C . - - \to C A = 0$

Hence, vector $- - \to B C$ is perpindicular to $- - \to C A .$

So, the given points are vertices of a right angled triangle.

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