MyQuestionIcon

3

You visited us 3 times! Enjoying our articles? Unlock Full Access!

Examples

Prove that fo...

Question

Prove that for any three vectors a, b, c [b + c c + a a + b] = 2 [abc].

Open in App

Solution

For any vector $∴ \to a, \to b$ and $\to c$ , prove that
$[\to a + \to b \to b + \to c \to c + \to a] = 2 [\to a \to b \to c]$
sowing LHS :-
$[\to a + \to b \to b + \to c \to c + \to a] = (\to a + \to b) . [(\to b + \to c) \times (\to c + \to a)]$
$[\to a \to b \to c = \to a . (\to b \times \to c]$
$= (\to a + \to b) . [(\to b \times \to c) + (\to b \times \to a) + (\to c \times \to c) + (\to c \times \to a)]$

$∵ [\to c \times \to c = | \to c | | \to c | sin O^n = o$
$= (\to a + \to b) . [(\to b \times \to c) + (\to b \times \to a) + o + (\to c \times \to a)]$
$= \to a . (\to b \times \to c) + \to b . (\to b \times \to c) + \to a . (\to b \times \to a) + \to b . (\to b \times \to a) + \to a . (\to c \times \to a) + \to b . (\to c \times \to a)$
$= [\to a, \to b, \to c] + [\to b \to b \to c] + [\to a \to b \to a] + [\to a \to c \to a] + [\to b \to c \to a]$

$∵ [\to b \to b \to c] = [\to c \to b \to b]$
$= \to c . (\to b \times \to b)$
As $(\to b \times \to b) = \to o$
$= \to c . o$
$= \to o$

$∵$ Using property
$[\to b \to b \to c] = 0$
$[\to a \to b \to a] = 0$
$[\to b \to b \to a] = 0$
$[\to a \to c \to a] = 0$

$= [\to a \to b \to c] + o + o + o + o + [\to b \to c \to a]$
$= [\to a \to b \to c] + [\to b \to c \to a]$
$= [\to a \to b \to c] + [\to a \to b \to c]$
$= 2 [\to a \to b \to c] = R . H . S$

Suggest Corrections

thumbs-up

0

Similar questions

Q. Prove that for any three vectors a, b, c

[b + c c + a a + b] = 2 [a b c]

Q. Prove that for any three vectors

\to a, \to b

\to c,

[(\to a + \to b) (\to b + \to c) (\to c + \to a)] = 2 [\to a \to b \to c] .

Q. For any three vectors

\to a, \to b

\to c

, prove that vectors

\to a - \to b, \to b - \to c

\to c - \to a

Q. For any three vectors

\to a, \to b

\to c

[\to a + \to b, \to b + \to c, \to c + \to a] = 2 [\to a \to b \to c]

. Hence prove that the vectors

\to a + \to b, \to b + \to c, \to c + \to a

are coplanar. If and only if

\to a, \to b, \to c

Q. For any three vectors

¯ ¯ ¯ a, ¯ ¯ b, ¯ ¯ c

[¯ ¯ ¯ a - ¯ ¯ b ¯ ¯ b - ¯ ¯ c ¯ ¯ c - ¯ ¯ ¯ a] = 0

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Integration

PHYSICS

Watch in App

explore_more_icon

Explore more

Standard XII Physics

Join BYJU'S Learning Program

CrossIcon