The length of the two diagonals of a parallelogram are:
[tex]\sqrt{2}\ \text{units\ and}\ \sqrt{10}\ \text{units}[/tex]
We know that if two vectors form the sides of a parallelogram then the two diagonals of the parallelogram are:
sum of the two vectors and difference of two vectors.
i.e. if u and v are two vectors such that they form the side of a parallelogram the,
u+v and u-v form the diagonal of a parallelogram.
Here we have:
u=<-1,1>
and v=<0,-2>
Hence,
u+v=<-1,1>+<0,-2>
i.e.
u+v=<-1+0,1-2>
i.e.
u+v=<-1,-1>
Also, the length of the diagonal will be:
||u+v||=||<-1,-1>||
[tex]||<-1,-1>||=\sqrt{(-1)^2+(-1)^2}\\\\i.e.\\\\||<-1,-1>||=\sqrt{2}[/tex]
Length of one diagonal is: √2 units
and the other diagonal is formed by the vector:
u-v
i.e.
u-v=<-1,1>-<0,-2>
i.e.
u-v=<-1-0,1-(-2)>
i.e.
u-v=<-1,3>
i.e. the length of the other diagonals is:
||u-v||=||<-1,3>||
Hence,
[tex]||<-1,3>||=\sqrt{(-1)^2+(3)^2}\\\\||<-1,3>||=\sqrt{1+9}\\\\i.e.\\\\||<-1,3>||=\sqrt{10}[/tex]
Hence, length of other diagonal is: √10 units.
The lengths of the two diagonals of the parallelogram are [tex]\sqrt{2}[/tex] and [tex]\sqrt{10}[/tex], respectively.
Vectorially speaking, the diagonals of the parallelogram can be found by means of the following vectorial formulas:
[tex]\vec {d}_{1} = \vec u + \vec v[/tex] (1)
[tex]\vec d_{2} = \vec u - \vec v[/tex] (2)
If we know that [tex]\vec u = \langle -1, 1 \rangle[/tex] and [tex]\vec v = \langle 0, -2 \rangle[/tex], then the vectors diagonal are:
[tex]\vec {d_{1}} = \langle -1, 1 \rangle + \langle 0, -2 \rangle[/tex]
[tex]\vec d_{1} = \langle -1, -1 \rangle[/tex]
[tex]\vec {d_{2}} = \langle -1, 1 \rangle - \langle 0, -2 \rangle[/tex]
[tex]\vec d_{2} = \langle -1, 3 \rangle[/tex]
And the lengths of the diagonals are calculated by Pythagorean theorem:
[tex]\|\vec {d}_{1}\| = \sqrt{(-1)^{2}+(-1)^{2}}[/tex]
[tex]\|\vec {d}_{1}\| = \sqrt{2}[/tex]
[tex]\|\vec {d}_{2}\| = \sqrt{(-1)^{2}+3^{2}}[/tex]
[tex]\|\vec {d}_2\| = \sqrt{10}[/tex]
The lengths of the two diagonals of the parallelogram are [tex]\sqrt{2}[/tex] and [tex]\sqrt{10}[/tex], respectively.
We kindly invite to check this question on parallelograms: https://brainly.com/question/1563728
