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Cross Product

If a⃗, b⃗, c⃗...

Question

If $\to a, \to b, \to c, \to a + \to b, \to b + \to c$ and $\to a + \to b + \to c$ are vectors of same magnitude, none of which is null vector, then which of the following is false

A

Angle between

\to b

and

\to c

will always be obtuse

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B

\to a - \to c

and

\to b

are mutually perpendicular vectors

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C

\to a + \to c

will be perpendicular to

\to b

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D

\to a

and

\to c

are mutually perpendicular vectors

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Solution

The correct option is C $\to a + \to c$ will be perpendicular to $\to b$
Let $| \to a | = | \to b | = | \to c | = | \to a + \to b | = | \to b + \to c | = | \to a + \to b + \to c | = k$
Let the angle between $\to a$ and $\to b$ is $θ_{1}$ , the angle between $\to b$ and $\to c$ is $θ_{2}$ , and the angle between $\to a$ and $\to c$ is $θ_{3}$

$| \to a + \to b |^{2} = k^{2} \Rightarrow k^{2} + k^{2} + 2 \to a . \to b = k^{2}$
$\Rightarrow \to a \cdot \to b = - \frac{k^{2}}{2} \Rightarrow θ_{1} = \frac{2 π}{3}$
Similarly $θ_{2} = \frac{2 π}{3}, θ_{3} = \frac{π}{2}$

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Similar questions

\to A = \to b \times \to c, \to B = \to c \times \to a, \to C = \to a \times \to b

, then the vectors

\to A \times (\to B \times \to C)

\to B \times (\to C \times \to A)

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