Why is a Parallelogram with Adjacent Sides Represented by this Vector Set?

[evinda]

Hello! (Wave)

Why does it hold that the parallelogram with adjacent sides the vectors $v_1$ and $v_2$ is this one: $\{ av_1+ b v_2: 0 \leq a, b \leq 1 \}$ ?

 

[jvicens]

Hello! This is a great question. The reason why the parallelogram with adjacent sides the vectors $v_1$ and $v_2$ is represented by the set $\{ av_1+ b v_2: 0 \leq a, b \leq 1 \}$ is because this set represents all possible linear combinations of $v_1$ and $v_2$ with coefficients $a$ and $b$ that range from 0 to 1. This means that any point within this set can be expressed as a combination of $v_1$ and $v_2$ with weights that fall within this range. Since a parallelogram is a quadrilateral with opposite sides parallel and equal in length, any point within this set can be connected to form the corners of a parallelogram. This is why the set $\{ av_1+ b v_2: 0 \leq a, b \leq 1 \}$ is a representation of the parallelogram with adjacent sides $v_1$ and $v_2$. I hope this helps clarify things! Let me know if you have any other questions.