Set 5 Problem number 4

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

What are the magnitude and angle of a vector whose x and y components are respectively -7.4 and 2.1?

Solution

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

• magnitude = `sqrt( ( -7.4) ^ 2 + ( 2.1) ^ 2) = 7.69.

The angle made by the vector with the x axis is one of two angles:

• angle = tan^-1 ( 2.1 m/s / (-7.4 m/s) ) = -15.85 degrees or (-15.85 + 180) degrees.

Since the x component of the vector is negative, the vector is in the second or third quadrant.

• Its angle with the positive x axis cannot therefore be equal to the -15.85 degrees found from the tan^-1 function, which is always a first- or fourth- fourth-quadrant angle.
• The angle is therefore (-15.85 + 180) degrees = 164.15 degrees, which is as required in the second or third quadrant.

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)