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Chapter 3: Problem 16

Find the magnitude of the vector \(34 \hat{\imath}+13 \hat{\jmath} \mathrm{m}\) and determine its angle to the \(x\) -axis.

Short Answer

Expert verified
The magnitude of the vector is approximately 36.4 m and the angle it makes with the x-axis is approximately 20.6 degrees.

Step by step solution

01

Find the Magnitude of the Vector

The magnitude of the vector \( V \) is given by: \( V \) = \( \sqrt{(34)^{2} + (13)^{2}}\)
02

Calculation of the Magnitude

By plugging the values and solving, we find that \( V \) = \( \sqrt{(1156)+ (169)} \) = \( \sqrt{1325} \) = 36.4 m
03

Find the Direction of the Vector

The angle \( \theta \) the vector makes with the x-axis can be found using the formula: \( \theta = tan^{-1} (\frac{y}{x}) \) where y and x are the respective j and i components of the vector.
04

Calculation of the Angle

By plugging the values into the formula, we find that \( \theta = tan^{-1} (\frac{13}{34}) \) = \( tan^{-1}(0.38) \) = 20.6 degrees

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You wish to row straight across a 63 -m-wide river. You can row at a steady \(1.3 \mathrm{m} / \mathrm{s}\) relative to the water, and the river flows at \(0.57 \mathrm{m} / \mathrm{s} .\) (a) What direction should you head? (b) How long will it take you to cross the river?

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