Question

1. Obtain the geometric interpretation of the Parallelogram law.
2. Prove the parallelogram law.

Short answer

1. The answer is stated below.
2. The answer is stated below.

Step-by-step solution

Draw the diagram of parallelogram.

(a)

Draw the diagram of parallelogram with\(a\) and\(b\) vectors as shown in Figure 1.

From figure 1, it is observed that\({\rm{a + b}}\) and\({\rm{a - b}}\) are the length of two diagonals.

It is also observed that\(a\)and\(b\)are the length of two sides, and\({\rm{ - a}}\) and \({\rm{ - b}}\)are the length of other two sides with opposite vector direction.

The Parallelogram law states that summation of square of the two diagonals is equal to the square of the four sides of parallelogram.

The mathematical expression is,

\(|{\rm{a}} + {\rm{b}}{|^2} + |{\rm{a}} - {\rm{b}}{|^2} = 2|{\rm{a}}{|^2} + 2|\;{\rm{b}}{|^2}\)

Hence, the required result is proved.

Formula used.

(b)

Consider the formula of\(|a + b{|^2}\).

\(|{\bf{a}} + {\bf{b}}{|^2} = |{\bf{a}}{|^2} + |{\bf{b}}{|^2} + 2({\bf{a}} \cdot {\bf{b}})\) …… (1)

Similarly consider the formula of\(|a - b{|^2}\).

\(|{\bf{a}} - {\bf{b}}{|^2} = |{\bf{a}}{|^2} + |{\bf{b}}{|^2} - 2({\bf{a}} \cdot {\bf{b}})\) …… (2)

Use the above formula for calculation.

Substitute equation (1) and (2) in equation\(|{\bf{a}} + {\bf{b}}| + |{\bf{a}} - {\bf{b}}|\)(1).

\(\begin{aligned}{l}|{\rm{a}}{|^2} + |{\rm{b}}{|^2} + 2({\rm{a}} \cdot {\rm{b}}) + |{\rm{a}}{|^2} + |{\rm{b}}{|^2} - 2({\rm{a}} \cdot {\rm{b}}) &= 2|{\rm{a}}{|^2} + 2|\;{\rm{b}}{|^2}\\2|{\rm{a}}{|^2} + 2|\;{\rm{b}}{|^2} + 2({\rm{a}} \cdot {\rm{b}}) - 2({\rm{a}} \cdot {\rm{b}}) &= 2|{\rm{a}}{|^2} + 2|\;{\rm{b}}{|^2}\\2|{\rm{a}}{|^2} + 2|\;{\rm{b}}{|^2} &= 2|{\rm{a}}{|^2} + 2|\;{\rm{b}}{|^2}\end{aligned}\)

Since the LHS and RHS are equal, the parallelogram law is proved.

Thus, the parallelogram law is\(|a + b{|^2} + |a - b{|^2} = 2|a{|^2} + 2|b{|^2}.\)