Set 5 Problem number 3

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

If a velocity vector has x and y components 3.5 meters/sec and 5.6 meters/sec, respectively, the what are the magnitude and angle the displacement?

Solution

The magnitude of the vector is found by the Pythagorean Theorem to be

• magnitude = `sqrt( ( 3.5 m/s) ^ 2 + ( 5.6 m/s) ^ 2) = 6.6 m/s.

The angle made by the vector with the x axis is

• angle = tan^-1 ( 5.6 m/s / ( 3.5 m/s) ) = 57.99 degrees

Since the components of the vector are both positive, the vector is in the first quadrant. Its angle with the positive x axis is therefore equal to the 57.99 degrees found from the tan^-1 function (the inverse tangent, found on a calculator from the inverse or 2d function key and the tan key).

Generalized Solution

The components vx and vy correspond to the sides of a right triangle whose hypotenuse is equal to the length or magnitude of the vector, and whose angle with the positive x axis is opposite to the y component and adjacent to the x component.

• The magnitude of the vector is therefore found from the Pythagorean Theorem to be v = `sqrt(vx^2 + vy^2).

The angle has a tangent (opposite side / adjacent side) equal to vy / vx.

• The angle is therefore the inverse tangent of this ratio:  angle = tan^-1(vy / vx).

Explanation in terms of Figure(s), Extension

The figure below depicts a vector with components vx and vy.

• By the Pythagorean Theorem the magnitude of the vector is `sqrt(vx^2 + vy^2).
• By the definition of the tangent, the angle of the vector with the positive x axis is tan^-1(vy / vx).

Figure(s)