SATHEE: Vectors 4 Question 3

Vectors 4 Question 3

3. Let $\hat{a}, \hat{b}$ and $\hat{c}$ be three unit vectors such that $\hat{a} \times (\hat{b} \times \hat{c}) = \frac{\sqrt{3}}{2} (\hat{b} + \hat{c})$ . If $\hat{b}$ is not parallel to $\hat{c}$ , then the angle between $\hat{a}$ and $\hat{b}$ is

(2016 Main)

(a) $\frac{3 π}{4}$

(b) $\frac{π}{2}$

(d) $\frac{5 π}{6}$

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Answer:

Correct Answer: 3. (d)

Solution:

Given, $| \hat{a} | = | \hat{b} | = | \hat{c} | = 1$

and $\hat{a} \times (\hat{b} \times \hat{c}) = \frac{\sqrt{3}}{2} (\hat{b} + \hat{c})$

Now, consider

$\hat{a} \times (\hat{b} \times \hat{c}) = \frac{\sqrt{3}}{2} (\hat{b} + \hat{c})$

$\Rightarrow (\hat{a} \cdot \hat{c}) \hat{b} - (\hat{a} \cdot \hat{b}) \hat{c} = \frac{\sqrt{3}}{2} \hat{b} + \frac{\sqrt{3}}{2} \hat{c}$

On comparing, we get

$\begin{aligned} \hat{a} \cdot \hat{b} = - \frac{\sqrt{3}}{2} \Rightarrow | \hat{a} | | \hat{b} | \cos θ = - \frac{\sqrt{3}}{2} \\ \Rightarrow & \cos θ = - \frac{\sqrt{3}}{2} [∵ | \hat{a} | = | \hat{b} | = 1] \\ \Rightarrow & \cos θ = \cos (π - \frac{π}{6}) \Rightarrow θ = \frac{5 π}{6} \end{aligned}$

Vectors 4 Question 3

3. Let $\hat{a}, \hat{b}$ and $\hat{c}$ be three unit vectors such that $\hat{a} \times (\hat{b} \times \hat{c}) = \frac{\sqrt{3}}{2} (\hat{b} + \hat{c})$ . If $\hat{b}$ is not parallel to $\hat{c}$ , then the angle between $\hat{a}$ and $\hat{b}$ is

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Vectors 4 Question 3

3. Let a^,b^ and c^ be three unit vectors such that a^×(b^×c^)=32(b^+c^). If b^ is not parallel to c^, then the angle between a^ and b^ is

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3. Let $\hat{a}, \hat{b}$ and $\hat{c}$ be three unit vectors such that $\hat{a} \times (\hat{b} \times \hat{c}) = \frac{\sqrt{3}}{2} (\hat{b} + \hat{c})$ . If $\hat{b}$ is not parallel to $\hat{c}$ , then the angle between $\hat{a}$ and $\hat{b}$ is