Let a = 2 i + j - 2 k , b = i + j and c = a × b . Let d be a vector such that | d - a | = 11 , | c × d | = 3 and the angle between c and d is 4 . Then a · d is equal to [2026]

Question

Let $\vec{a} = 2 \hat{i} + \hat{j} - 2 \hat{k}, \vec{b} = \hat{i} + \hat{j}$ and $\vec{c} = \vec{a} \times \vec{b} .$ Let $\vec{d}$ be a vector such that $| \vec{d} - \vec{a} |$ $= \sqrt{11},$ $| \vec{c} \times \vec{d} |$ $= 3$ and the angle between $\vec{c}$ and $\vec{d}$ is $\frac{π}{4}$ . Then $\vec{a} \cdot \vec{d}$ is equal to [2026]

Smart Achievers · Accepted Answer

(3)

$\vec{c} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & - 2 \\ 1 & 1 & 0 \end{matrix} |$

$\vec{c} = 2 \hat{i} + 2 \hat{k} + \hat{k}, | \vec{c} | = 3$

$| \vec{c} \times \vec{d} |$ $= 3$

$\vec{c}$ $| \vec{d} |$ $\sin \frac{π}{4} = 3 \Rightarrow$ $| \vec{d} |$ $= \sqrt{2}$

$| \vec{d} - \vec{a} |$ $= \sqrt{11}$

$\Rightarrow | \vec{a} |^{2} + | \vec{d} |^{2} - 2 \vec{a} \cdot \vec{d} = 11$

$9 + 2 - 2 \vec{a} \cdot \vec{d} = 11$

$\vec{a} \cdot \vec{d} = 0$

Solution

1 3

2 1

3 0

4 11

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Solution

Q. Let a→=2i^+j^-2k^, b→=i^+j^ and c→=a→×b→. Let d→ be a vector such that |d→-a→|=11, |c→×d→|=3 and the angle between c→ and d→ is π4. Then a→·d→ is equal to [2026]

1 3

2 1

3 0

4 11

Ans. (3) c→=|i^j^k^21-2110| c→=2i^+2k^+k^, |c→|=3 |c→×d→|=3 |c→||d→|sinπ4=3⇒|d→|=2 |d→-a→|=11 ⇒|a→|2+|d→|2-2a→·d→=11 9+2-2a→·d→=11 a→·d→=0

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