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Magnitude and direction of the vectors

The formula for direction can be calculated using the inverse tangent on the component ratios:

If v_x represents horizontal component and v_y represents vertical component., then the direction of formula: \theta = \tan^{-1} \frac{v_y}{v_x}

The Pythagorean theorem can be used to measure the speed magnitude at any instant based on the components available at the time. The magnitude of the speed v = \sqrt{v_x^2 + v_y^2}

Step 1: Create a suitable figure based on the details provided.

Speed of the rain V_r = 35 m/s

Speed of the wind V_w = 12 m/s

Wind direction = east to west

Rain direction = vertically downward (north to south)


Step 2: Calculating the resultant velocity vector magnitude and

Horizontal component, V_r = 35 m/s   and

Vertical component, V_w = 12 m/s

Magnitude of the resultant vector |V_R|= \sqrt{V_r^2 + V_m^2}

|V_R| = \sqrt{35^2 + 12^2} = 37 m/s


Step 3: Determine the resultant velocity vector direction

Direction of resultant \tan \theta = components ratio

\tan \theta = \frac{opp}{adj} = \frac{12}{35}

\theta = \tan^{-1} (\frac{12}{35}) = 18.9 \approx 19\degree

Thus, in the vertical plane at an angle of approximately 19\degree   the boy must keep his umbrella vertically east.