Let- a and c are unit vectors and - b- - 4- The angle between a and c is cos -114- If b -2 c - a- then - is equal to

The correct option is D 3, 4 Given: #160;| a #160;| =1,| c #160;| =1 and #160;| b #160;| =4. #160;⇒| a . c #160;| =1.1. 1 ...

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Byju's Answer

Standard XII

Mathematics

Applications of Dot Product

Let, a and ...

Question

Let, $\to a$ and $\to c$ are unit vectors and $∣ ∣ ∣ \to b ∣ ∣ ∣ = 4$ . The angle between $\to a$ and $\to c$ is ${cos}^{- 1} (\frac{1}{4})$ . If $\to b - 2 \to c = λ \to a$ , then $λ$ is equal to

3, 4

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- 3, 4

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3, - 4

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\frac{1}{4}, \frac{3}{4}

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Solution

The correct option is D $3, - 4$
Given: $∣ ∣ \to a ∣ ∣ = 1, ∣ ∣ \to c ∣ ∣ = 1$ and $∣ ∣ ∣ \to b ∣ ∣ ∣ = 4.$
$\Rightarrow ∣ ∣ \to a . \to c ∣ ∣ = 1.1. \frac{1}{4} = \frac{1}{4}$
Now, $\to b - 2 \to c = λ \to a \Rightarrow \to a . \to b - 2 \to a . \to c = λ a^{2}$
$\Rightarrow \to a . \to b - 2. \frac{1}{4} = λ$ $\Rightarrow \to a . \to b = λ + \frac{1}{2}$
Again, $\to b - 2 \to c = λ \to a \Rightarrow \to b . \to b - 2 \to b . \to c = λ \to b . \to a$
$\Rightarrow 16 - 2 \to b . \to c = λ (λ + \frac{1}{2})$ $\Rightarrow \to b . \to c = 8 - \frac{λ^{2}}{2} - \frac{λ}{4}$
Also, $\to b - 2 \to c = λ \to a \Rightarrow \to b . \to c - 2 \to c . \to c = λ \to b . \to a$
$\Rightarrow 8 - \frac{λ^{2}}{2} - \frac{λ}{4} - 2 (1) = λ (\frac{1}{4})$
$\Rightarrow λ^{2} + λ - 12 = 0$
$∴ λ = - 4, 3.$

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Let, →a and →c are unit vectors and ∣∣∣→b∣∣∣=4. The angle between →a and →c is cos−1(14). If →b−2→c=λ→a, then λ is equal to

Let, $\to a$ and $\to c$ are unit vectors and $∣ ∣ ∣ \to b ∣ ∣ ∣ = 4$ . The angle between $\to a$ and $\to c$ is ${cos}^{- 1} (\frac{1}{4})$ . If $\to b - 2 \to c = λ \to a$ , then $λ$ is equal to