If A-a-2 b-3 c-B-2 a-3 b-2 c-C-8 a-13b-Then A-B- and A-C- are collinear

A⃗B⃗=O⃗B⃗ O⃗A⃗ =( 2a⃗+3b⃗+2c⃗) (a⃗ 2b⃗+3c⃗) = 3a⃗+5b⃗ c⃗ A⃗C⃗=O⃗C⃗ O⃗A⃗ =( 8a⃗+13b⃗) (a⃗ 2b⃗+3c⃗) = 9a⃗+15b⃗ 3c⃗ =3( 3a⃗+5b⃗ c⃗) =3A⃗B⃗ ⇒A⃗C⃗=3A⃗B⃗ ...

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Byju's Answer

Standard XIII

Mathematics

Test for Collinearity of Points

If A⃗=a⃗-2 b⃗...

Question

If $\to A = (\to a - 2 \to b + 3 \to c)$
$\to B = (- 2 \to a + 3 \to b + 2 \to c)$
$\to C = (- 8 \to a + 13 \to b)$
Then $\to A B$ and $\to A C$ are collinear

True

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False

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Solution

The correct option is A True
$\to A B = \to O B - \to O A = (- 2 \to a + 3 \to b + 2 \to c) - (\to a - 2 \to b + 3 \to c) = - 3 \to a + 5 \to b - \to c \to A C = \to O C - \to O A = (- 8 \to a + 13 \to b) - (\to a - 2 \to b + 3 \to c) = - 9 \to a + 15 \to b - 3 \to c = 3 (- 3 \to a + 5 \to b - \to c) = 3 \to A B \Rightarrow \to A C = 3 \to A B S o, \to A C i s p a r a l l e l t o \to A B But they have a common point A S o \to A C a n d \to A B a r e c o l l i n e a r$

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If →A=(→a−2→b+3→c) →B=(−2→a+3→b+2→c) →C=(−8→a+13→b) Then →AB and →AC are collinear

If $\to A = (\to a - 2 \to b + 3 \to c)$
$\to B = (- 2 \to a + 3 \to b + 2 \to c)$
$\to C = (- 8 \to a + 13 \to b)$
Then $\to A B$ and $\to A C$ are collinear