Set 5 Problem number 2

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

A displacement vector has length 5 and is directed at an angle with the positive x axis of 112 degrees.  What are the displacements in the x and y direction which would give the same net displacement?

Solution

The displacement vector is completely equivalent to the combination of its x and y displacements, as calculated:

• The x component is 5 cos( 112 deg) = -1.88.
• The y component is 5 sin( 112 deg) = 4.63.

Note that the x and y components may be positive or negative, depending on the quadrant of the vector.  Recalling that angles are measured in the counterclockwise direction from the positive x axis, we see that:

• A vector with angle between 90 and 180 degrees is in the second quadrant and has a negative x component and a positive y component.
• A vector with angle between 180 and 270 degrees is in the third quadrant and has a negative x component and a negative y component.
• A vector with angle between 270 and 360 degrees is in the fourth quadrant and has a positive x component and a negative y component.

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)