and also perpendicular to each other?
how can we prove that ...please Help me
and also perpendicular to each other?
how can we prove that ...please Help me
Two vectors are orthogonal if and only if their dot product is 0. Can you use that fact in your proof?
A hint. We have a general arrangement like this
If we let $\vec{BA} = \begin{pmatrix} a \\ b \end{pmatrix}$ then what will the vector $\vec{DA}$ be? How do we represent a vector perpendicular to a given vector?
What about $\vec{CA}$? And $\vec{BD}$? Can they be written in terms of $\vec{BA}$ and $\vec{DA}$?
How do we determine when two vectors are perpendicular?
Let |A|
be Euclidean length of vector A
Now as per Pythagoras theorem: For two orthogonal vector - say A
& B
- in 2D,
|A+B| = sqrt(|A|^2 + |B|^2)
Now, let a
, b
, c
, and d
are the corner point of given square in clockwise direction and ab
, bc
, cd
, and da
are vectors which represents the sides of square.
So, |ab| = |bc| = |cd| = |da|
So, dimension of diagonal:
{Dimension_of ac
or ca
} = |ac|
= sqrt(|ab|^2 + |bc|^2)
= sqrt(|ab|^2 + |ad|^2)
= |bd|
= {Dimension_of bd
or db
}
Above argument is correct for 2D. and Square is a planar figure, so we can apply these rules.
So, this proves that, two diagonals of a square are equal in dimension.
Done!