For any two vectors a- and b- which of the following are ture -A- - a - b -2- a - b 2- a -2- b -2B- -1- a - b -2- a - b - a - b -2- a -2- b -2C- a - b - a - a -2 b - a - b aD- a - b - a - a -2 b - a - b a

The correct options are A |→a×→b|^2 + (→a.→b)^2 = |→a|^2|→b|^2 B |1 →a . →b|^2 + |→a + →b + →a×→b|^2 = |→a|^2 |→b|^2 C (→a×→b) ×→a = |→a|^2 →b |(→a. →b)→a D →a ...

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Byju's Answer

Standard XII

Mathematics

Applications of Cross Product

For any two v...

Question

For any two vectors $\to a$ and $\to b$ , which of the following are ture ?

| \to a \times \to b |^{2} + (\to a . \to b)^{2} = | \to a |^{2} | \to b |^{2}

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| 1 - \to a . \to b |^{2} + | \to a + \to b + \to a \times \to b |^{2} = | \to a |^{2} | \to b |^{2}

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(\to a \times \to b) \times \to a = | \to a |^{2} \to b - | (\to a . \to b) \to a

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\to a \times (\to b \times \to a) = | \to a |^{2} \to b - | (\to a . \to b) \to a

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Solution

The correct options are
A $| \to a \times \to b |^{2} + (\to a . \to b)^{2} = | \to a |^{2} | \to b |^{2}$
B $| 1 - \to a . \to b |^{2} + | \to a + \to b + \to a \times \to b |^{2} = | \to a |^{2} | \to b |^{2}$
C $(\to a \times \to b) \times \to a = | \to a |^{2} \to b - | (\to a . \to b) \to a$
D $\to a \times (\to b \times \to a) = | \to a |^{2} \to b - | (\to a . \to b) \to a$
$| \to a \times \to b |^{2} = [\begin{matrix} \to a . \to a & \to a . \to b \to b . \to a & \to b . \to b \end{matrix}]$

$= | \to a |^{2} | \to b |^{2} - (\to a . \to b)^{2}$

Therefore

$| \to a \times \to b |^{2} + (\to a . \to b)^{2} = | \to a |^{2} | \to b |^{2}$

We have,

$| 1 - \to a . \to b |^{2} + | \to a + \to b + \to a \times \to b |^{2}$

$1 - 2 (\to a . \to b) + (\to a . \to b)^{2} + | \to a |^{2} + | \to b |^{2} + (\to a . \to b) + | \to a \times \to b |^{2}$

$[∵ \to a . (\to a . \to b) = \to b . (\to a \times \to b) = 0] = 1 + | \to a |^{2} | \to b |^{2} c o s^{2} θ + | \to a |^{2} + | \to b |^{2} + | \to a |^{2} | \to b |^{2} s i n^{2} θ = 1 + | \to a |^{2} + | \to b |^{2} + | \to a |^{2} | \to b |^{2} (c o s^{2} θ + s i n^{2} θ) = (1 + | \to a |^{2}) (1 + | \to b |^{2})$

$(\to a \times \to b) \times \to a = (\to a . \to a) \to b - (\to a . \to b) \to a$

Now,

$\to a \times (\to b \times \to a) = (\to a . \to a) \to b - (\to a . \to b) \to a$

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