00:01
In this question, we want to verify that these four points are the vertices of a parallelogram and the fine area.
00:08
Now for a parallelogram, if this is a, b, c, d, c, will be parallel.
00:24
And ad and b c will be parallel to.
00:29
And not only that, they will have, a .b will have the same line at d .c.
00:34
And ad will have the same line as b.
00:37
And for its area, the area will be taking the vector ab, cross it with ad.
00:47
The magnitude will be the area.
00:49
So the area of the parallelogram, abcd, will be the magnitude of ab, cross its adjacent vector ad.
01:08
So let's verify first if abcd are vertices or parallelogram.
01:13
Now notice that ab cd are written as coordinate form.
01:21
So let's express it as column vector form.
01:24
Column vector will be oa and that will be column vector like this.
01:32
O b vector oc and od.
01:49
Okay, so let's find ab first, vector ab.
01:56
Factor ab will be ob minus oa and that will be one to be one 2, 3.
02:10
Let's find vector dc.
02:15
There'll be oc minus od and that will also be 1, 2, 3.
02:25
So you can see that ab is parallel to dc, right? you can see that they are parallel because you can see that the vectors are actually the same.
02:37
So there are many 2 is also the same.
02:40
So ab, which is the length, is also equals to dc.
02:45
Now the other pair, ad, over here so ad will be od minus oa you get 540 and we will have bc this one over here bc will be oc minus ob and again we get 540 so you can see that ad is parallel to bc okay so these two sets are parallel and not only that you can see their same vector so they will have the same magnitude.
03:34
So this goes to show that abcd actually forms a parallelogram.
03:40
So implies abcd is a parallelogram.
03:48
So we're shown.
03:50
Okay, so we want to find the area now of the parallelogram...