00:01
So we need to show that area of parallelograms interform diagonal is half d1 times d2 if diagonals are spanned by vector u and b.
00:09
So let's draw a parallelogram first and here we can see let's this is our diagonal.
00:24
So this is our diagonal.
00:30
Let's assume this is d1 and hence it has vector u and similarly.
00:36
So take this direction in the direction and we have diagonal d2 which is vector v so now we can take or name this vector this is our origin point so this is a vector this is b vector and since it is parallel this will also be b vector having the same magnitude in direction so positive positive and this will be a vector okay so now to find the area of this parallel we can look into the triangles here so if we look to the triangles okay this one this one so if we draw height so area of this triangle is area of triangle so it is half times base which is a so magnitude of a bar times height which will be sine theta if this is our angle so sine theta of b right since we apply technometry sine theta so that will be opposite which is the height divided by adjacent sorry ipytonous which is our epitonous so from this we get height as magnitude of b b sine theta so we can write here sine theta so this is our area of triangle and since this also this triangle is the same same as the first one so we can write area of parallelogram will be double of this so that is a times b sine theta and this is nothing but a cross product of a b vector so now we have our relation in terms of ab we need to convert in terms of u and v to convert that into diagonal form so let's see here so a bar plus b bar gives us u bar so here a bar is sorry a bar plus b bar is u bar and similarly, if you look at the v, which is a bar minus b bar, right? if you go in the negative direction, we reach to vector v.
02:46
So a bar minus b bar is vector v bar.
02:50
If we add this to vector, we get 2a bar is equal to u bar plus v bar, and therefore a bar is half times u bar plus v bar.
03:00
Similarly, if we subtract these two vectors, so we'll get 2b bar as this.
03:07
U bar minus v bar therefore b bar is half times u bar minus v bar now we need a cross b so we'll take a cross b so a parallelogram a cross b and this will be equal to half times half since it's a scalar value so that will be multiplied here and rest will multiply so u plus v cross u minus v so this will be 1 by 4...