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 the value of -a-b-c- is

The correct option is C 32 a =2∧ #160; i +∧ #160; j 2∧ #160; k b =∧ #160; i +∧ #160; j a × b : | ∧ #160; i ...

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Byju's Answer

Standard XII

Mathematics

Applications of Cross Product

Let a = 2î ...

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 the value of $| (\to a \times \to b) \times \to c |$ is

\frac{2}{3}

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\frac{3}{2}

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2

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Solution

The correct option is C $\frac{3}{2}$
$¯ ¯ ¯ a = 2 \land i + \land j - 2 \land k$
$¯ ¯ b = \land i + \land j$
$¯ ¯ ¯ a \times ¯ ¯ b :$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} \land i & \land j & \land k 2 & 1 & - 2 1 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$⟹ \land i (0 + 2) - \land j (2) + \land k (2 - 1)$
$⟹ 2 \land i - 2 \land j + \land k$
$∣ ∣ ¯ ¯ ¯ a \times ¯ ¯ b ∣ ∣ = \sqrt{2^{2} + 2^{2} + 1^{2}} = 3$
$∣ ∣ ¯ ¯ c - ¯ ¯ ¯ a ∣ ∣ = 2 \sqrt{2}$
$⟹ \sqrt{{∣ ∣ ¯ ¯ c ∣ ∣}^{2} + {∣ ∣ ¯ ¯ ¯ a ∣ ∣}^{2} - 2 ¯ ¯ ¯ a . ¯ ¯ c} = 2 \sqrt{2}$
$⟹ \sqrt{{∣ ∣ ¯ ¯ c ∣ ∣}^{2} + {∣ ∣ ¯ ¯ ¯ a ∣ ∣}^{2} - 2 ∣ ∣ ¯ ¯ c ∣ ∣} = 2 \sqrt{2} - (i)$
$(∵ ¯ ¯ ¯ a . ¯ ¯ c = ∣ ∣ ¯ ¯ c ∣ ∣)$
$∣ ∣ (¯ ¯ ¯ a \times ¯ ¯ b) \times ¯ ¯ c ∣ ∣ = ∣ ∣ ¯ ¯ ¯ a \times ¯ ¯ b ∣ ∣ . ∣ ∣ ¯ ¯ c ∣ ∣ s i n 30^{\circ}$
$= 3. ∣ ∣ ¯ ¯ c ∣ ∣ s i n 30^{\circ}$
$= \frac{3 ∣ ∣ ¯ ¯ c ∣ ∣}{2} - (i i)$
$¯ ¯ ¯ a = 2 \land i + \land j - 2 \land k$
$∣ ∣ ¯ ¯ ¯ a ∣ ∣ = \sqrt{2^{2} + 1 + 2^{2}} = 3$
Putting $∣ ∣ ¯ ¯ ¯ a ∣ ∣ i n (i)$
$⟹ \sqrt{{∣ ∣ ¯ ¯ c ∣ ∣}^{2} + 9 - 2 ∣ ∣ ¯ ¯ c ∣ ∣} = 2 \sqrt{2}$
Squaring:
$⟹ {∣ ∣ ¯ ¯ c ∣ ∣}^{2} - 2 ∣ ∣ ¯ ¯ c ∣ ∣ + 9 = 8$
$⟹ {∣ ∣ ¯ ¯ c ∣ ∣}^{2} - 2 ∣ ∣ ¯ ¯ c ∣ ∣ + 1 = 0$
$⟹ {∣ ∣ ¯ ¯ c ∣ ∣}^{2} - ∣ ∣ ¯ ¯ c ∣ ∣ - ∣ ∣ ¯ ¯ c ∣ ∣ + 1 = 0$
$⟹ {(∣ ∣ ¯ ¯ c ∣ ∣ - 1)}^{2} = 0$
$⟹ ∣ ∣ ¯ ¯ c ∣ ∣ = 1$
Putting $∣ ∣ ¯ ¯ c ∣ ∣ i n (i i)$
$⟹ ∣ ∣ (¯ ¯ ¯ a \times ¯ ¯ b) \times ¯ ¯ c ∣ ∣ = \frac{3}{2} ∣ ∣ ¯ ¯ c ∣ ∣ = \frac{3}{2}$

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Similar questions

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

and the angle between

(\to a \times \to b)

and

\to c

30^{0}

, then

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

is equal to

Q. Let

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

and

\to b =^i +^k

. 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}

, then

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

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

\to a \times \to b

and

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, then

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

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 |

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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

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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 the value of |(→a×→b)×→c| is

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 the value of $| (\to a \times \to b) \times \to c |$ is