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General Solution of cos theta = cos alpha

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Q.150. If the position vectors of three points are $\vec{a} - 2 \vec{b} + 3 \vec{c}, 2 \vec{a} + 3 \vec{b} - 4 \vec{c} a n d - 7 \vec{a} + 10 \vec{c}$ , then the three points are
(a) collinear
(b) non-collinear
(c) coplanar
(d) None of these

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Solution

$L e t p o i s t i o n v e c t o r o f P (\vec{p}) = \vec{a} - 2 \vec{b} + 3 \vec{c} Q (\vec{q}) = 2 \vec{a} + 3 \vec{b} - 4 \vec{c} R (\vec{r}) = - 7 \vec{a} + 10 \vec{c} L e t u s f i r s t a s s u m e t h a t t h e s e p o in t s a r e c o l l i n e a r . T h e n P Q R l i e i n a s t r i a g h t l i n e . V e e c t o r \vec{P Q} m u s t b e p a r a l l e l t o \vec{Q R} \vec{P Q} = \vec{q} - \vec{p} = (2 \vec{a} + 3 \vec{b} - 4 \vec{c}) - (\vec{a} - 2 \vec{b} + 3 \vec{c}) = \vec{a} + 5 \vec{b} - 7 \vec{c} \vec{Q R} = \vec{r} - \vec{q} = (- 7 \vec{a} + 10 \vec{c}) - (2 \vec{a} + 3 \vec{b} - 4 \vec{c}) = - 9 \vec{a} - 3 \vec{b} - 14 \vec{c} \vec{P Q} d o e s n' t l o o k s p a r a l l e l t o \vec{Q R} . H e n c e w e c a n n o t s a y t h e y a r e c o l l i n e a r . A l s o w e c a n n o t c o m m e n t h a t t h e y a r e n o n - c o l l i n e a r a s \vec{a}, \vec{b} a n d \vec{c} m i g h t t a k e v a l u e s s u c h t h a t \vec{P Q} b e c o m e s p a r a l l e l t o \vec{P R} . S i n c e w e d o n t k n o w \vec{a}, \vec{b} a n d \vec{c} h e n c e w e c a n n o t c o m m m e n t B u t w e k n o w a n y t h r e e p o i n t s a l w a y s l i e o n a p l a n e . S o w e c a n d e f i n i t e l y s a y t h a t t h e y a r e c o - p l a n a r .$

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Similar questions

Q. If the position vectors of three points are

\to a - 2 \to b + 3 \to c, 2 \to a + 3 \to b + 4 \to c, - 7 \to b + 10 \to c

, then the three points are

Q. Show that the points A, B, C with position vectors

\vec{a} - 2 \vec{b} + 3 \vec{c}, 2 \vec{a} + 3 \vec{b} - 4 \vec{c}

- 7 \vec{b} + 10 \vec{c}

State True or False.

The following position vectors are collinear.

a - 2 b + 3 c

2 a + 3 b - 4 c

- 7 b + 10 c

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Q. The three points A, B, C with position vectors

- 2 a + 3 b + 5 c, a + 2 b + 3 c, 7 a - c

Q. There are three points with position vectors

- 2 a + 3 b + 5 c, a + 2 b + 3 c

7 a - c

. What is the relation between the three points?

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General Solution of cos theta = cos alpha

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