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Dot Product of Two Vectors

a⃗ and c⃗ a...

Question

$\to a$ and $\to c$ are unit vectors and $| \to b | = 4$ with $\to a \times \to b = 2 \to a \times \to c$ . The angle between $\to a$ and $\to c$ is ${cos}^{- 1} (\frac{1}{4})$ . Then $\to b - 2 \to c = λ \to a$ , if $λ$ is

A

3

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B

\frac{1}{4}

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C

- 4

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D

- \frac{1}{4}

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Solution

The correct options are
A $3$
D $- 4$
$\to a \times (\to b - 2 \to c) = 0$
$\to b - 2 \to c | | \to a$
$\to b - 2 \to c = k \to a$ ---(1)
$| \to b | = | k \to a + 2 \to c |$
$4 = \sqrt{k^{2} | \to a |^{2} + 4 | \to c |^{2} + 4 k \to c \to a}$
$16 = k^{2} + 4 + k$
$= k^{2} + k - 12$
$= (K + 4) (K - 3)$
$K = - 4, 3$

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