Let a- - 2- - - - 2k- and b- - - - - If c- is a vector such that a- c- - -c- -c- - a- - 2-2 and the angle between a-b- and c- is 30- - then -a-b-c- is equal to

In this question, vector c⃗ is not given, therefore, we cannot apply the formulae of (a⃗×b⃗×c⃗) (vector triple product). Now, |(a⃗×b⃗)×c⃗| = |a⃗×b⃗||c⃗|sin 30^∘ ...

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Byju's Answer

Standard XIII

Mathematics

Cross Product of Two Vectors

Let a⃗ = 2î +...

Question

Let $\to a = 2^i +^j - 2^k a n d \to b =^i +^j .$ If $\to c$ is a vector such that $\to a . \to c = | \to c |, | \to c - \to a | = 2 \sqrt{2}$ and the angle between $(\to a \times \to b) a n d \to c i s 30^{\circ}, t h e n | (\to a \times \to b) \times \to c |$ is equal to

\frac{2}{3}

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\frac{3}{2}

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2

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3

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Solution

The correct option is B $\frac{3}{2}$
In this question, vector $\to c$ is not given, therefore, we cannot apply the formulae of $(\to a \times \to b \times \to c)$ (vector triple product).
Now, $| (\to a \times \to b) \times \to c | = | \to a \times \to b | | \to c | sin 30^{\circ} . . . (i) Again, | \to a \times \to b | = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 1 & - 2 1 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 2^i - 2^j +^k \Rightarrow | \to a \times \to b | = \sqrt{2^{2} + (- 2)^{2} + 1} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 Since, | \to c - \to a | = 2 \sqrt{2} [given] \Rightarrow | \to c - \to a |^{2} = 8 \Rightarrow (\to c - \to a) . (\to c - \to a) = 8 \Rightarrow \to c . \to c - \to c . \to a - \to a . \to c + \to a . \to a = 8 \Rightarrow | \to c |^{2} + | \to a |^{2} - 2 | \to c | = 8 \Rightarrow | \to c |^{2} + 9 - 2 | \to c | = 8 \Rightarrow | \to c |^{2} - 2 | \to c | + 1 = 0 \Rightarrow (| \to c | - 1)^{2} = 0 \Rightarrow | \to c | = 1 From, Eq . (i), | (\to a \times \to b) \times \to c | = (3) (1) . (\frac{1}{2}) = \frac{3}{2}$

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Similar questions

Q. Let

\to a = 2^i +^j - 2^k, \to b =^i +^j

. If

\to c

is a vector such that

\to a . \to c = | \to c |

and

| \to c - \to a | = 2 \sqrt{2}

and angle between

\to a \times \to b

and

\to c

30^{o}

. Then,

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

equals

Q. Let

\to a = 2^i +^j - 2^k

and

\to b =^i +^j

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\to c

be a vector such that

| \to c - \to a | = 3 | (\to a \times \to b) \times \to c | = 3

and the angle between

\to c

and

\to a \times \to b

30^{o}

. Then

\to a . \to c

is equal to

Q. Let

\to a = 2^i +^j - 2^k

and

\to b =^i +^j

. If

\to c

is a vector such that

\to a . \to c = | \to c |

| \to c - \to a | = 2 \sqrt{2}

and the angle between

(\to a \times \to b)

and

\to c

30^{0}

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∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ =

Q. Let

\to a = 2^i +^j - 2^k

and

\to b =^i +^j

. Let

\to c

be a vector such that

| \to c - \to a | = 3

| (\to a \times \to b) \times \to c | = 3

and the angle between

\to c

and

\to a \times \to b

30^{o}

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\to a \cdot \to c

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Q. Let

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

and

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Let →a=2^i+^j−2^k and →b=^i+^j. If →c is a vector such that →a.→c=|→c|, |→c−→a|=2√2 and the angle between (→a×→b) and →c is 30∘ ,then |(→a×→b)×→c| is equal to

Let $\to a = 2^i +^j - 2^k a n d \to b =^i +^j .$ If $\to c$ is a vector such that $\to a . \to c = | \to c |, | \to c - \to a | = 2 \sqrt{2}$ and the angle between $(\to a \times \to b) a n d \to c i s 30^{\circ}, t h e n | (\to a \times \to b) \times \to c |$ is equal to