SATHEE: Vectors 1 Question 2

Vectors 1 Question 2

2. If a unit vector a makes angles $\frac{π}{3}$ with $\hat{i}, \frac{π}{4}$ with $\hat{j}$ and $θ \in (0, π)$ with $\hat{k}$ , then a value of $θ$ is

(2019 Main, 9 April II)

(a) $\frac{5 π}{6}$

(b) $\frac{π}{4}$

(c) $\frac{5 π}{12}$

(d) $\frac{2 π}{3}$

Show Answer

Answer:

Correct Answer: 2. (d)

Solution:

Given unit vector a makes an angle $\frac{π}{3}$ with $\hat{i}, \frac{π}{4}$ with $\hat{j}$ and $θ \in (0, π)$ with $\hat{k}$ .

Now, we know that $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$ , where $α, β, γ$ are angles made by the vectors

with respectively $\hat{i}, \hat{j}$ and $\hat{k}$ .

$∴ \cos^{2} (\frac{π}{3}) + \cos^{2} (\frac{π}{4}) + \cos^{2} θ = 1$

$\Rightarrow \frac{1}{4} + \frac{1}{2} + \cos^{2} θ = 1 \Rightarrow \cos^{2} θ = \frac{1}{4} \Rightarrow \cos θ = \pm \frac{1}{2}$

$\Rightarrow \cos θ = \cos (\frac{π}{3})$ or $\cos (\frac{2 π}{3}) \Rightarrow θ = \frac{π}{3}$ or $\frac{2 π}{3}$

So, $θ$ is $\frac{2 π}{3}$ , according to options.

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