Let a -2i-j-2k and b -i-j- It 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 - is equal to

The correct option is B #160; 3 / 2 We have #160; a × b = i amp; j amp; k 2 amp; 1 amp; 2 1 amp; 1 amp; 0 =2i 2j+k ...

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Byju's Answer

Standard XII

Mathematics

Applications of Cross Product

Let a =2i+j...

Question

Let $\to a = 2 i + j - 2 k$ and $\to b = i + j$ . It $\to c$ 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 $∣ ∣ ∣ (\to a \times \to b) \times \to c ∣ ∣ ∣$ 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 $\frac{3}{2}$
We have $\to a \times \to b = ∣ ∣ ∣ ∣ \begin{matrix} i & j & k 2 & 1 & - 2 1 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ = 2 i - 2 j + k$
$\Rightarrow ∣ ∣ ∣ \to a \times \to b ∣ ∣ ∣ = \sqrt{4 + 4 + 1} = 3$
Now, ${∣ ∣ \to c - \to a ∣ ∣}^{2} = {(2 \sqrt{2})}^{2}$
$\Rightarrow {∣ ∣ \to c ∣ ∣}^{2} + {∣ ∣ \to a ∣ ∣}^{2} - 2 \to c . \to a = 8 \Rightarrow {∣ ∣ \to c ∣ ∣}^{2} + 4 + 1 + 4 - 2 ∣ ∣ \to c ∣ ∣ = 8 \Rightarrow {∣ ∣ \to c ∣ ∣}^{2} - 2 ∣ ∣ \to c ∣ ∣ + 1 = 0 \Rightarrow {(∣ ∣ \to c ∣ ∣ - 1)}^{2} = 0 \Rightarrow ∣ ∣ \to c ∣ ∣ = 1$
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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Similar questions

Q. Let

a = 2 i + j - 2 k

and

b = i + j

. If

c

is a vector such that

a . c = | c |, | c - a | = 2 \sqrt{2}

and the angle between

a \times b

and

c

30^{0}

then

| (a \times b) \times c |

is equal to

Q. Let

a = 2 i + j - 2 k

and

b = i + j

. If

c

is a vector such that

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

and the angle between

(a \times b)

and

c

30^{o}

, then

| (a \times b) \times c |

is equal to

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 \to i + \to j - 2 \to k

and

\to b = \to i + \to 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^{\circ}

, 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 +^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

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

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Let →a=2i+j−2k and →b=i+j. It →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∣∣∣ is equal to