Chapter 5, Problem 95A

To determine

The conditions under which the Pythagorean theorem, rather than the law of cosines, be used to find the magnitude of a resultant vector.

Answer to Problem 95A

If the two vectors to be added are at right angles to each other than the Pythagorean theorem can be used to find the magnitude of a resultant vector.

Explanation of Solution

Introduction:

Pythagorean Theorem is one of the most significant relations in mathematics.It describes the correlation among the three sides of a right triangle. Pythagorean theorem states that the square of the length of the hypotenuse side of a right-angled triangle is equivalent to the sum of squares of length of the other two sides of thatright-angled triangle.

Consider a right-angled triangle ABC as shown in the figure 1. Let the length of the sides AB and BC be x and y respectively. Length of the hypotenuse side AC be z . Then, the Pythagorean Theorem for this right-angled triangle can be written as,

  $x^2+y^2=z^2$

Or, $z=\sqrt{x^2+y^2}$

    Figure:1

Pythagorean theorem can be used to find the magnitude of a resultant vector if and only if the two vectors to be added are at right angles to each other. If $\overrightarrow{A}$ and $\overrightarrow{B}$ are the two displacement vectors that are at right angles to each other, then the resultant vector $\overrightarrow{R}$ can be represented as shown in the figure 2.

    Figure 2

Magnitude of the resultant vector can be written as,

  $R=\sqrt{A^2+B^2}$ .