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 300- then - a-b- -c- -

The correct option is B #160; 3 2 We have, #160;a⃗×#160; b=î amp; ĵ amp; k̂ 2 amp; 1 amp; 1 1 amp; 1 amp; 0 =2î 2ĵ+k̂⇒| a ...

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Byju's Answer

Standard XII

Mathematics

Condition for Coplanarity of Four Points

Let a⃗=2î+ĵ...

Question

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 $\to c$ is $30^{0}$ , then $∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ =$

\frac{2}{3}

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\frac{3}{2}

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2

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Solution

The correct option is B $\frac{3}{2}$
We have, $\to a \times \to b = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 1 & - 1 1 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 2^i - 2^j +^k \Rightarrow ∣ ∣ \to a \times \to b ∣ ∣ = \sqrt{4 + 4 + 1} = 3$

Also, ${| \to c - \to a |}^{2} = {(2 \sqrt{2})}^{2} \Rightarrow {| \to c |}^{2} + {| \to a |}^{2} - 2 \to c . \to a = 8$

$\Rightarrow {| \to c |}^{2} + 4 + 1 + 4 - 2 | \to c | = 8 \Rightarrow {| \to c |}^{2} - 2 | \to c | + 1 = 0$

$\Rightarrow {({| \to c |}^{2} - 1)}^{2} = 0 \Rightarrow | \to c | = 1$

Now, $∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ = ∣ ∣ \to a \times \to b ∣ ∣ | \to c | sin (30^{0}) = (3) (1) (\frac{1}{2}) = \frac{3}{2}$

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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 300, then ∣∣(→a×→b)×→c∣∣=

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 $\to c$ is $30^{0}$ , then $∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ =$