The three points whose position vectors are -2 -3 k- 3 -4 -7 k- and -3 -2 -5 k-A- form the vertices of an equilateral triangleB- form the vertices of a right angled triangleC- are collinearD- form the vertices of an isosceles triangle-

The correct option is C are collinearIf A, B, C are the given points respectively, then O⃗A=î+2ĵ+3k̂, O⃗B=3î+4ĵ+7k̂, O⃗C= 3î 2ĵ 5k̂, A⃗B=O⃗B O⃗A=2î+2ĵ+4k̂, A⃗ ...

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Standard IX

Mathematics

Collinear Vectors

The three poi...

Question

The three points whose position vectors are $^i + 2^j + 3^k, 3^i + 4^j + 7^k a n d - 3^i - 2^j - 5^k$

form the vertices of an equilateral triangle

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form the vertices of a right angled triangle

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are collinear

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form the vertices of an isosceles triangle.

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Solution

The correct option is C are collinear
If A, B, C are the given points respectively, then
$\to O A =^i + 2^j + 3^k, \to O B = 3^i + 4^j + 7^k, \to O C = - 3^i - 2^j - 5^k, \to A B = \to O B - \to O A = 2^i + 2^j + 4^k, \to A C = \to O C - \to O A = - 4^i - 4^j - 8^k = - 2 \to A B$
$∴ \to A B, \to A C$ are collinear $\Rightarrow A, B, C$ are collinear.

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The three points whose position vectors are ^i+2^j+3^k,3^i+4^j+7^k and−3^i−2^j−5^k

The correct option is C are collinearIf A, B, C are the given points respectively, then →OA=^i+2^j+3^k,→OB=3^i+4^j+7^k,→OC=−3^i−2^j−5^k,→AB=→OB−→OA=2^i+2^j+4^k,→AC=→OC−→OA=−4^i−4^j−8^k=−2→AB ∴→AB,→AC are collinear ⇒A,B,C are collinear.

The three points whose position vectors are $^i + 2^j + 3^k, 3^i + 4^j + 7^k a n d - 3^i - 2^j - 5^k$