If a - b and c are three non - coplanar vectors- then a - b - c - a - b - a - c equalsA- 0B- -a-b-c-C- 2-a-a-c-D- -a-b-c-

The correct option is D [a⃗b⃗c⃗]We have,(a⃗+b⃗)× (a⃗+c⃗)=a⃗×c⃗+b⃗×a⃗+b⃗×c⃗ = a⃗× b +b⃗×c⃗ c⃗×a⃗ ∴ (a⃗+b⃗+c⃗). { (a⃗+b⃗)× (a⃗+c⃗) } =(a⃗+b⃗+c⃗).( a⃗×b⃗+b⃗×c⃗ c ...

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Byju's Answer

Standard IX

Mathematics

Test for Coplanarity

If a , b and ...

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

\to 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, $(\to a + \to b) \times (\to a + \to c) = \to a \times \to c + \to b \times \to a + \to b \times \to c$
$= - \to a \times b + \to b \times \to c - \to c \times \to a ∴ (\to a + \to b + \to c) . {(\to a + \to b) \times (\to a + \to c)}$
$= (\to a + \to b + \to c) . (- \to a \times \to b + \to b \times \to c - \to c \times \to a)$
$= [\to a \to a \to b] + [\to a \to b \to c] - [\to a \to c \to a] - [\to b \to a \to b] + [\to b \to b \to c] - [\to b \to c \to a] - [\to c \to a \to b] + [\to c \to b \to c] - [\to c \to c \to a] = [\to a \to b \to c] - [\to b \to c \to a] - [\to c \to a \to b] = - [\to a \to b \to c]$

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Similar questions

If $\to a, \to b$ and $\to c$ are three non-coplanar vectors, then $(\to a + \to b + \to c) \cdot [(\to a + \to b) \times (\to a + \to c)]$ equals

Q. If

a, b, c

are three non-coplanar vectors, then

(a + b + c) \cdot ((a + b) \times (a + c))

is equal to

Q. If

\vec{a,} \vec{b,} \vec{c}

are three non-coplanar vectors, then

(\vec{a} + \vec{b} + \vec{c}) . [(\vec{a} + \vec{b}) \times (\vec{a} + \vec{c})]

equals

(a) 0

(b)

[\vec{a} \vec{b} \vec{c}]

(c)

2 [\vec{a} \vec{b} \vec{c}]

(d)

- [\vec{a} \vec{b} \vec{c}]

Q. If

¯ ¯ ¯ a, ¯ ¯ b, ¯ ¯ c

are three non-coplanar vectors,

¯ ¯ ¯ p, ¯ ¯ ¯ q, ¯ ¯ ¯ r

are non-zero vectors such that

\to p = \frac{\to b \times \to c}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}, \to q = \frac{\to c \times \to a}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}, \to r = \frac{\to a \times \to b}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}

, then the value of the expression

(\to a + \to b + \to c) . (\to p + \to q + \to r)

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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