Set 5 Problem number 1

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

A force vector has length 6 and is directed at an angle with the positive x axis of 27 degrees.  What are the lengths of the force vectors in the x and y direction which would have the same effect as this force?

Solution

The vector is completely equivalent in its effect to the combined effect of its x and y components.

• The x component is 6 cos( 27 deg) = 5.34.
• The y component is 6 sin( 27 deg) = 2.72.

Generalized Solution

The components of a vector of magnitude v which makes angle `theta with the positive x axis, as measured counterclockwise from the axis, are v<sub>x</sub> = v cos(`theta) and v_y = v sin(`theta).

• These components can be understood geometrically from a sketch in which the magnitude of the vector its represented by its length, with its angle measured with respect the positive x axis of a standard set of coordinate axes.
• The sine and cosine (abbreviated sin and cos as above) of an angle are simply the ratios between the y component of the vector and its length, and between the x component and the length. They are easily found on a calculator.

Explanation in terms of Figure(s), Extension

The figure below depicts a vector of length v making angle `theta with the positive x axis, as measured counterclockwise from that axis.

• The components v_x and v_y can be seen as sides of a right triangle formed by the vector and by projection lines parallel to the coordinate axes.

Figure(s)