Mathematics 10 — Exercise 6.3 · Q7

Full question

Show by vectors that these vertices form a parallelogram.

Step-by-step solution

1. Vector from the first point to the second is endpoint minus starting point.

   $$5\times î+2\times ĵ-(î+2\times ĵ)$$

2. Subtract the matching starting coordinate from the endpoint coordinate.

   $$5-1=4$$

3. Subtract the matching starting coordinate from the endpoint coordinate.

   $$2-2=0$$

4. Write the two calculated components as a vector.

   $$4\times î+0\times ĵ$$

5. Vector from the first point to the second is endpoint minus starting point.

   $$7\times î+6\times ĵ-(3\times î+6\times ĵ)$$

6. Subtract the matching starting coordinate from the endpoint coordinate.

   $$7-3=4$$

7. Subtract the matching starting coordinate from the endpoint coordinate.

   $$6-6=0$$

8. Write the two calculated components as a vector.

   $$4\times î+0\times ĵ$$

9. Vector from the first point to the second is endpoint minus starting point.

   $$7\times î+6\times ĵ-(5\times î+2\times ĵ)$$

10. Subtract the matching starting coordinate from the endpoint coordinate.

    $$7-5=2$$

11. Subtract the matching starting coordinate from the endpoint coordinate.

    $$6-2=4$$

12. Write the two calculated components as a vector.

    $$2\times î+4\times ĵ$$

13. Vector from the first point to the second is endpoint minus starting point.

    $$3\times î+6\times ĵ-(î+2\times ĵ)$$

14. Subtract the matching starting coordinate from the endpoint coordinate.

    $$3-1=2$$

15. Subtract the matching starting coordinate from the endpoint coordinate.

    $$6-2=4$$

16. Write the two calculated components as a vector.

    $$2\times î+4\times ĵ$$

17. The two pairs of opposite directed sides have equal components. Hence opposite sides are equal and parallel, forming a parallelogram.

    $$(\left[\begin{array}{c}4\\ 0\end{array}\right],\left[\begin{array}{c}4\\ 0\end{array}\right],\left[\begin{array}{c}2\\ 4\end{array}\right],\left[\begin{array}{c}2\\ 4\end{array}\right])$$

Answer

Parallelogram.