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Chapter 16: Problem 1512

If unit vector \(\underline{a}\) and \(\underline{b}\) form an angle of \((\pi / 6)\) and \((2 \pi / 3)\) with positive direction of \(\mathrm{x}\) -axis respectively, then \(|\underline{\mathrm{a}}+\underline{\mathrm{b}}|=\) (A) \(\sqrt{(2 / 3)}\) (B) 2 (C) \(\sqrt{2}\) (D) \(\sqrt{3}\)

Short Answer

Expert verified
\( |\underline{a} + \underline{b}| = \sqrt{\frac{2}{3}}\)

Step by step solution

01

Determine the components of unit vectors

Since the magnitude of a unit vector is 1, we can use basic trigonometry to find the components of \(\underline{a}\) and \(\underline{b}\). For any vector and its angle with the x-axis, the components are: \(a_x = |\underline{a}| \cos{\theta_a}\) \(a_y = |\underline{a}| \sin{\theta_a}\) \(b_x = |\underline{b}| \cos{\theta_b}\) \(b_y = |\underline{b}| \sin{\theta_b}\) In this case, we have: \(a_x = \cos{\frac{\pi}{6}}\) \(a_y = \sin{\frac{\pi}{6}}\) Furthermore, \(b_x = \cos{\frac{2\pi}{3}}\) \(b_y = \sin{\frac{2\pi}{3}}\)
02

Add both vectors component-wise

To obtain the sum of these vectors, we add their respective components: \((a_x + b_x) = \cos{\frac{\pi}{6}} + \cos{\frac{2\pi}{3}}\) \((a_y + b_y) = \sin{\frac{\pi}{6}} + \sin{\frac{2\pi}{3}}\)
03

Calculate the magnitude of the sum vector

Now, we can find the magnitude of the sum vector using the calculated components: \( |\underline{a} + \underline{b}| = \sqrt{(a_x + b_x)^2 + (a_y + b_y)^2} \) \(= \sqrt{(\cos{\frac{\pi}{6}} + \cos{\frac{2\pi}{3}})^2 + (\sin{\frac{\pi}{6}} + \sin{\frac{2\pi}{3}})^2} \) After simplifying: \(= \sqrt{\left(\frac{\sqrt{3}}{2} - \frac{1}{2}\right)^2 + \left(\frac{1}{2} + \frac{\sqrt{3}}{2}\right)^2} \) \(= \sqrt{\frac{3}{4}} \) Comparing with the given choices: \( |\underline{a} + \underline{b}| = \boxed{\text{(A) } \sqrt{\frac{2}{3}}} \)

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Most popular questions from this chapter

Equation of line passes through \(\mathrm{A}(-2,4,7)\) and direction \((5,-9,12)\) is (A) \(x=-2+\overline{k 5, y}=4-9 k, z=7-12 k, k \in R\) (B) \(\mathrm{x}=-2+\mathrm{k} 5, \mathrm{y}=4-9 \mathrm{k}, \mathrm{z}=7+12 \mathrm{k}, \mathrm{k} \in \mathrm{R}\) (C) \(x=2+5 k, y=4-9 k, z=7+12 k, k \in R\) (D) None of these

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