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Chapter 1: Problem 32

Unit vector parallel to the resultant of vectors \(\vec{A}=4 \hat{i}-3 \hat{j}\) and \(\vec{B}=8 \hat{i}+8 \hat{j}\) will be (A) \(\frac{24 \hat{i}+5 \hat{j}}{13}\) (B) \(\frac{12 \hat{i}+5 \hat{j}}{13}\) (C) \(\frac{6 \hat{i}+5 \hat{j}}{13}\) (D) None of these

Short Answer

Expert verified
The unit vector parallel to the resultant of vectors \(\vec{A}\) and \(\vec{B}\) is \(\boxed{\frac{12\hat{i} + 5\hat{j}}{13}}\), which matches option (B).

Step by step solution

01

Calculate the resultant vector

First, we will sum the given vectors, \(\vec{A}\) and \(\vec{B}\), to get the resultant vector \(\vec{R}\). We have, \(\vec{A}=4 \hat{i}-3 \hat{j}\) and \(\vec{B}=8 \hat{i}+8 \hat{j}\). Let's add the vectors component-wise as follows: \(\vec{R} = \vec{A} + \vec{B} = \left( 4\hat{i} - 3\hat{j} \right) + \left( 8\hat{i} + 8\hat{j} \right) = \left( 4 + 8 \right) \hat{i} + \left(-3 + 8 \right) \hat{j} = 12\hat{i} + 5\hat{j}.\) So, the resultant vector, \(\vec{R}=12\hat{i}+5\hat{j}\).
02

Calculate the magnitude of the resultant vector

Now, let's find the magnitude of the resultant vector \(\vec{R}\). The magnitude is given by: \(|\vec{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{(12^2) + (5^2)} = \sqrt{144 + 25}=\sqrt{169}=13\)
03

Find the unit vector parallel to the resultant vector

Finally, we will divide the resultant vector by its magnitude to find the unit vector parallel to it, denoted as \(\hat{R}\): \(\hat{R} = \frac{\vec{R}}{|\vec{R}|} = \frac{12\hat{i} + 5\hat{j}}{13}\). The unit vector parallel to the resultant of vectors \(\vec{A}\) and \(\vec{B}\) is \(\boxed{\frac{12\hat{i} + 5\hat{j}}{13}}\), which matches option (B).

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