Set 5 Problem number 5

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

What are x and y the components of the displacement vector obtained when we add the two following displacement vectors:

• vector A, with x and y components 2.9 meters and 5 meters, and
• vector B whose x and y components are 8.5 meters and -5.5 meters?

What are the magnitude and angle of the resultant vector?

Solution

The x component is simply the sum 2.9 meters + 8.5 meters = 11.4 meters of the x components of the vectors being added.

The y component is similarly the sum 5 meters + -5.5 meters = -.5 meters of the y components of the added vectors.

The magnitude of the resultant vector, by the Pythagorean Theorem, is therefore

• magnitude = `sqrt( ( 11.4 meters )² + ( -.5 meters )²) = 11.41 meters .

The angle of the resultant vector to the x axis is

• angle = tan^-1( -.5 meters /( 11.4 meters )) = -2.52 degrees.

Since the x component of this vector is positive, this is the correct angle.

Generalized Solution

Vectors A and B, with their x and y components indicated, are shown on the figure below. The sum of the two x components will be the x component of the resultant, and the sum of the two y components will be the y component of the resultant:

R_x = A_x + B_x

and

R_y = A_y + B_y.

We find the magnitude and angle of R, using the Pythagorean Theorem and the inverse tangent:

| R | = `sqrt(R_x² + R_y²)

and

`theta = tan^-1(R_y / R_x).

Explanation in terms of Figure(s), Extension

The figure below shows two vectors A and B, and their components A_x, A_y, B_x and B_y.

• The y components A_y and B_y are seen to add up to R_y, as the x components A_x and B_x are seen to at up to R_x.
• The magnitude and angle of the vector R are obtained by using the Pythagorean Theorem and the inverse tangent in the usual manner.

Figure(s)