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Multiplication with Vectors

Let v = i × ...

Question

Let
$\to v = \to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to j \times \to k))))))))$
Then the value of $∣ ∣ ∣ ∣ \to v ∣ ∣ ∣ ∣$ is

A

0

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B

1

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C

\sqrt{3}

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D

3

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Solution

The correct option is C $1$
$\to v = \to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to j \times \to k))))))))$

$\to i \times \to j = \to k, \to j \times \to k = \to i, \to k \times \to i = j$

$\to j \times \to i = - \to k, \to k \times \to j = - \to i, \to i \times \to k = - \to j$

$∴ \to v = \to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to k \times (\to i \times (\to j \times \to i)))))))$

$\to v = \to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to k \times (\to i \times - \to k))))))$

$\to v = \to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to k \times \to j)))))$

$\to v = \to i \times (\to j \times (\to k \times (\to i \times (\to j \times - \to i))))$

$\to v = \to i \times (\to j \times (\to k \times (\to i \times \to k)))$

$\to v = \to i \times (\to j \times (\to k \times - \to j))$

$\to v = \to i \times (\to j \times \to i)$

$\to v = \to i \times - \to k$

$\to v = \to j$

$| | \to v | | = 1$

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Similar questions

\to v = \to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to k \times (\to i \times (\to j \times (\to j \times \to k))))))))

.
Then the value of

| | \to v | |

{\to V}_{1} =^i +^j - 2^k, {\to V}_{2} =^i - 2^j +^k, {\to V}_{3} = - 2^i + 2^j +^k

are three vectors. Let

\to V

be a vector such that it can be expressed as a linear combination of

{\to V}_{1}

{\to V}_{2}

\to V {\to V}_{3} = 0

and the projection of the vector

\to V

^i -^j +^k

6 \sqrt{3}

\to V = λ (^i + 3^j - 4^k)

then find the absolute value of

λ

Q. The value of

\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}),

is
(a) 0
(b) −1
(c) 1
(d) 3

Let $\to a = i + 2 j + k, \to b = i - j + k a n d \to c = i + j - k,$ a vector in the plane $\to a a n d \to b$ whose projection on $\to c i s \frac{1}{\sqrt{3}} i s . . . . . . . .$

\to a =^i +^j +^k

\to b =^i -^j +^k

\to c =^i -^j -^k

be three vectors. A vector

\to v

in the plane of

\to a

\to b

, whose projection on

\to c

\frac{1}{\sqrt{3}}

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