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Geometric Progression

For any vecto...

Question

For any vector $\to a$ , the value of $(\to a \times^i)^{2} + (\to a \times^j)^{2} + (\to a \times^k)^{2}$ is equal to

A

3 {\to a}^{2}

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B

{\to a}^{2}

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C

2 {\to a}^{2}

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D

4 {\to a}^{2}

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Solution

The correct option is C $2 {\to a}^{2}$
Let $\to a = x^i + y^j + z^k$
$\Rightarrow \to a \times^i = z^j - y^k$
$=> (\to a \times^i)^{2} = y^{2} + z^{2}$
Similarly $(\to a \times^j)^{2} = x^{2} + z^{2}$
and $(\to a \times^k)^{2} = x^{2} + y^{2}$
$∴ (\to a \times^i)^{2} + (\to a \times {^j}^{2} + (\to a \times^k)^{2} = 2 (x^{2} + y^{2} + z^{2}) = 2 {\to a}^{2}$

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Similar questions

{\to A}_{1}

{\to A}_{2}

are two vectors such that

| {\to A}_{1} | = 3, | {\to A}_{2} | = 5

| {\to A}_{1} + {\to A}_{2} | = 5

(2 {\to A}_{1} + 3 {\to A}_{2}) \cdot (2 {\to A}_{1} - 2 {\to A}_{2})

| {\to A}_{1} | = 2

| {\to A}_{2} | = 3

| {\to A}_{1} + {\to A}_{2} | = 3

. Find the value of

({\to A}_{1} + 2 {\to A}_{2}) \cdot (3 {\to A}_{1} - 4 {\to A}_{2})

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Q. For any vectors

\to a

(\to a \times^i)^{2} + (\to a \times^j)^{2} + (a \times^k)^{2}

Q. For any three vectors

\to a, \to b, \to c

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

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$\to a, t h e v a l u e o f (\to a \times^i)^{2} + (\to a \times^j)^{2} + (\to a \times^k)^{2} i s (a) {\to a}^{2} (b) 3 \to a (c) 4 {\to a}^{2} (d) 2 {\to a}^{2}$

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