If A-B- and C- are three non-coplanar vectors- then A-B-C- -A-B-A-C- -equals

The correct option is D [ #160;A #160;B #160;C] #160;We have (A + #160;B + #160;C).[(A + #160;B) × (A + #160;C)] = (A + #160;B + ...

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Byju's Answer

Standard XII

Mathematics

Applications of Dot Product

If A⃗,B⃗ an...

Question

If $\to A, \to B$ and $\to C$ are three non-coplanar vectors, then
$(\to A + \to B + \to C)$ . $[(\to A + \to B) \times (\to A + \to C)]$ equals

0

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[\to A \to B \to C]

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2 [\to A \to B \to C]

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- [\to A \to B \to C]

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Solution

The correct option is D $- [\to A \to B \to C]$
We have
$(¯ ¯¯ ¯ A + ¯ ¯¯ ¯ B + ¯ ¯¯ ¯ C) . [(¯ ¯¯ ¯ A + ¯ ¯¯ ¯ B) \times (¯ ¯¯ ¯ A + ¯ ¯¯ ¯ C)]$

$= (¯ ¯¯ ¯ A + ¯ ¯¯ ¯ B + ¯ ¯¯ ¯ C) . [(¯ ¯¯ ¯ A \times ¯ ¯¯ ¯ C) + (¯ ¯¯ ¯ B \times ¯ ¯¯ ¯ A) + (¯ ¯¯ ¯ B \times ¯ ¯¯ ¯ C)]$
$= [¯ ¯¯ ¯ A ¯ ¯¯ ¯ B ¯ ¯¯ ¯ C] - [¯ ¯¯ ¯ A ¯ ¯¯ ¯ B ¯ ¯¯ ¯ C] - [¯ ¯¯ ¯ A ¯ ¯¯ ¯ B ¯ ¯¯ ¯ C]$
$= - [¯ ¯¯ ¯ A ¯ ¯¯ ¯ B ¯ ¯¯ ¯ C]$

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Similar questions

Q. If

\to a, \to b

and

\to c

are three non-coplanar vectors, prove that

[\to a + \to b + \to c \to a + \to b \to a + \to c] = - [\to a \to b \to c]

Q. If

\to a \times (\to b \times \to c) = (\to a \times \to b) \times \to c

for non-coplanar vectors

\to a

\to b

\to c

then

Q. Let

\to a, \to b, \to c

be three vectors such that

\to a \neq 0

and

\to a \times \to b = 2 \to a \times \to c, | \to a | = | \to c | = 1, | \to b | = 4

and

| \to b \times \to c | = \sqrt{15}

. If

\to b - 2 \to c = λ \to a

, then

λ

is equal to

Q. Vectors

\to a, \to b

and

\to c

are such that

\to a + \to b + \to c = \to 0

and

| \to a | = 3, | \to b = 5

and

| \to c | = 7

. Find the angle between

\to a

and

\to b .

Q. If

\to a, \to b

and

\to c

are perpendicular to

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

and

\to a + \to b

respectively and if

| \to a + \to b | = 6, | \to b + \to c | = 8

and

| \to c + \to a | = 10

, the

| \to a + \to b + \to c |

is equal to

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If →A,→B and →C are three non-coplanar vectors, then (→A+→B+→C).[(→A+→B)×(→A+→C)]equals

If $\to A, \to B$ and $\to C$ are three non-coplanar vectors, then
$(\to A + \to B + \to C)$ . $[(\to A + \to B) \times (\to A + \to C)]$ equals