Chapter 14.1, Problem 9OE

To determine

To explain: Whether the statement “A parallelogram $ABCD$ have the sides $\overrightarrow{2DX}+\overrightarrow{BC}=\overrightarrow{DC}$ .” is true or false.

  

Answer to Problem 9OE

The statement “A parallelogram $ABCD$ have the sides $\overrightarrow{2DX}+\overrightarrow{BC}=\overrightarrow{DC}$ .” is true.

Explanation of Solution

Given information:

The statement “A parallelogram $ABCD$ have the sides $\overrightarrow{2DX}+\overrightarrow{BC}=\overrightarrow{DC}$ .”

Consider the provided statement “A parallelogram $ABCD$ have the sides $\overrightarrow{2DX}+\overrightarrow{BC}=\overrightarrow{DC}$ .”

Parallelogram has a property that opposite sides of a parallelogram are parallel to each other.

Parallelogram has four sides and four angles, sometimes sides and angles are equal and sometimes they may different.

Each diagonal bisects each other and each diagonal divides the quadrilateral into two equal congruent triangles.

Thus the, $\overrightarrow{2DX}+\overrightarrow{BC}=\overrightarrow{DC}$ ,

Here,the two vectors are of the same length and same direction.

  $DX=XB$

By substituting the above values in the equations.

  $\begin{array}{c}\overrightarrow{2DX}+\overrightarrow{BC}=\overrightarrow{DC}\\ 2DX=DX+XB\\ =DB\end{array}$

Now the two vectors form a triangle that show the sum vector.

The sum angle is that angle made by when two vectors make a triangle in sequence, the third side that closing the triangle, in the opposite direction of the sequence represents the sum or resultant of two vectors .

Now substitute to get,

  $\begin{array}{c}DB+BC=DC\\ \overrightarrow{2DX}+\overrightarrow{BC}=\overrightarrow{DC}\end{array}$

So the given expression $\overrightarrow{2DX}+\overrightarrow{BC}=\overrightarrow{DC}$ is true.

Thus, the statement “A parallelogram $ABCD$ have the sides $\overrightarrow{2DX}+\overrightarrow{BC}=\overrightarrow{DC}$ .” is true.