# Vector components question

• Coco12
In summary, a cyclist traveling west at 5.6m/s encounters a northeast wind blowing at 10m/s. To find the effective speed of the tailwind, the components of the velocities are added together using Pythagorean theorem. The resultant speed is 1.5m/s due to the westward velocity of the cyclist being represented as a negative value.

## Homework Statement

A cyclist head due west on a straight road at 5.6m/s. A northeast wind is blowing at 10m/s. What is the effective speed of the tailwind?(resultant)

Cos 45 10
Sin 45 10

## The Attempt at a Solution

Basically I broke it down into its x and y components. Then added them together to the Rx and Ry and used Pythagorean to find the resultant. I just have one question though: when adding the x components for the wind and the cyclist speed. Would the Rx be -5.6m/s + 7.1( 7.1 is derived from cos 45 degrees *10) = 1.5? Or would the 5.6 be positive?

Coco12 said:
Would the Rx be -5.6m/s + 7.1( 7.1 is derived from cos 45 degrees *10) = 1.5? Or would the 5.6 be positive?

To help you answer your question, it might help to write out explicit expressions for the vectors representing the cyclist and the wind velocities.

CAF123 said:
To help you answer your question, it might help to write out explicit expressions for the vectors representing the cyclist and the wind velocities.

I did. Rx=ax+bx
Ry=ay +by

I'm just wondering since it's 5.6 m/s due west which on a Cartesian plane would be a negative x whether I would incorporate the negative when trying to find rx

Coco12 said:
I did. Rx=ax+bx
Ry=ay +by

I'm just wondering since it's 5.6 m/s due west which on a Cartesian plane would be a negative x whether I would incorporate the negative when trying to find rx
Yes, you can represent the velocity vector of the cyclist as ##\vec{C} = -5.6 \hat{x}## and that of the wind as ##\vec{W} = (10 \cos 45)\hat{x} + (10 \sin 45) \hat{y}##. Now, as you said, just add components to get the resultant.

From a more conceptual point of view, imagine a game of tug of war. Person A pulls to left with force 5.6N and person B to right with force 10cos45 ≈ 7.1 N. The person pulling to right wins, but only marginally. (winning by 7.1 - 5.6, not 7.1 + 5.6)

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