Assertion -If three points P- Q and R have position vectors a- b- and c- respectively- and 2 a- - 3 b- - 5 c- - 0- then the points P- Q and R must be collinear- Reason- If for three points A- B and C- A-B- -A-C- then points A- B and C must be collinear-

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Byju's Answer

Standard XII

Mathematics

Test for Collinearity of Vectors

Assertion :If...

Question

Assertion :If three points $P, Q$ and $R$ have position vectors $\to a, \to b$ and $\to c$ , respectively, and $2 \to a + 3 \to b - 5 \to c = 0$ , then the points $P, Q$ and $R$ must be collinear. Reason: If for three points $A, B$ and $C$ , $\to A B = λ \to A C$ , then points $A, B$ and $C$ must be collinear.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
$2 \to a + 3 \to b - 5 \to c = 0$

$\Rightarrow 3 (\to b - \to a) = 5 (\to c - \to a) \Rightarrow \to A B = \frac{5}{3} \to A C$
Hence, $\to A B$ and $\to A C$ must be parallel since there is a common point $A$ . The points $A, B$ and $C$ must be collinear.

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Assertion :If three points P,Q and R have position vectors →a,→b and →c, respectively, and 2→a+3→b−5→c=0, then the points P,Q and R must be collinear. Reason: If for three points A,B and C, →AB=λ→AC, then points A,B and C must be collinear.

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion2→a+3→b−5→c=0⇒3(→b−→a)=5(→c−→a)⇒→AB=53→ACHence, →AB and →AC must be parallel since there is a common point A. The points A,B and C must be collinear.