Let a -2 -2 k- b - - If c is a vector such that a - c - c - a - c -2 -2 and the angle between a - b and c is -6-thenA- - a - b - c -3-2B- -c-1- -a-b- -c-2-3D- - c -4

The correct options are A |(a×b )×c| = 32 B |c| = 1|(a×b )×c| = |a×b ||c| sinπ6 a×b = î amp; ĵ amp; k̂ 2 amp; 1 amp; 2 1 amp; 1 amp; 0 ⇒a×b ...

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Byju's Answer

Standard XII

Mathematics

Cross Product

Let a =2 î+ĵ-...

Question

Let $\to a = 2^i +^j - 2^k, \to b =^i +^j$ . If $\to c$ is a vector such that $\to a \cdot \to c = | \to c |, | \to a - \to c | = 2 \sqrt{2}$ and the angle between $\to a \times \to b$ and $\to c$ is $\frac{π}{6}$ , then

∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ = \frac{3}{2}

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| \to c | = 1

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∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ = \frac{2}{3}

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| \to c | = 4

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Solution

The correct options are
A $∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ = \frac{3}{2}$
B $| \to c | = 1$
$∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ = | \to a \times \to b | | \to c | sin \frac{π}{6} \to a \times \to b = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 1 & - 2 1 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣ \Rightarrow \to a \times \to b = 2^i - 2^j +^k \Rightarrow | \to a \times \to b | = 3$

Using $| \to a - \to c | = 2 \sqrt{2}$
$| \to a - \to c |^{2} = 8 \Rightarrow | \to a |^{2} + | \to c |^{2} - 2 \to a \cdot \to c = 8 \Rightarrow 9 + | \to c |^{2} - 2 | \to c | = 8 \Rightarrow | \to c | = 1$

So,
$∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ = | \to a \times \to b | | \to c | sin \frac{π}{6} \Rightarrow ∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ = 3 \times 1 \times \frac{1}{2} \Rightarrow ∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ = \frac{3}{2}$

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Similar questions

Q. Let

\to a = 2^i +^j - 2^k a n d \to b =^i +^j .

\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

Q. Let

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

and

\to b =^i +^j .

\to c

is a vector such that

\to a \cdot \to c = ∣ ∣ \to c ∣ ∣,

∣ ∣ \to c - \to a ∣ ∣ = 2 \sqrt{2}

and the angle between

(\to a \times \to b)

and

\to c

\frac{π}{6},

then the value of

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

Q. If

\to a^{'} =^i +^j, \to b^{'} =^i +^j + 2^k

and

\to c^{'} = 2^i +^j -^k

. Then altitude of the parallelopiped formed by the vectors

\to a, \to b, \to c

having base formed by

\to b

and

\to c

(\to a, \to b, \to c

and

{\to a}^{'}, {\to b}^{'}, {\to c}^{'}

are reciprocal system of vectors)

Q. Let

\to a = 2^i +^j - 2^k a n d \to b =^i +^j .

\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

Q. Let

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

. If

^c

is a vector such that

\to a . \to c = | \to c |, | \to a + \to c | = 2 \sqrt{3}

and angle between

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and

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

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Let →a=2^i+^j−2^k,→b=^i+^j. If →c is a vector such that →a⋅→c=|→c|,|→a−→c|=2√2 and the angle between →a×→b and →c is π6, then

Let $\to a = 2^i +^j - 2^k, \to b =^i +^j$ . If $\to c$ is a vector such that $\to a \cdot \to c = | \to c |, | \to a - \to c | = 2 \sqrt{2}$ and the angle between $\to a \times \to b$ and $\to c$ is $\frac{π}{6}$ , then