In Exercises 41 and 42, (a) verify that the points are the vertices of a parallelogram, (b) find

step 1 Hello, hope you're doing well. So the problem gives us these four points. And the first part of

In Exercises 41 and 42, (a) verify that the points are the...

Key Concepts

Dot Product for Checking Perpendicularity

The dot product of two vectors helps determine if they are perpendicular. When the dot product is zero, the vectors are orthogonal. This concept is used to verify whether the parallelogram is a rectangle by checking if adjacent sides are at right angles.

Vector Cross Product for Area Calculation

The area of a parallelogram can be computed by taking the cross product of two adjacent side vectors. The magnitude of the resulting vector gives the area, serving as a powerful method in three-dimensional space to calculate areas directly from vector components.

Vector Representation

This concept involves representing points in space as vectors and using vector subtraction to obtain directional vectors between points. In problems like this, converting points into vectors enables us to analyze the geometric properties of the figure using algebraic operations.

Properties of a Parallelogram

A parallelogram is a quadrilateral where opposite sides are parallel and equal in magnitude. Key properties include the fact that the vectors representing opposite sides are equal, and the diagonals bisect each other. Verifying these properties algebraically confirms the figure is a parallelogram.

Transcript

00:01 Hello, hope you're doing well.

00:03 So the problem gives us these four points.

00:05 And the first part of the problem asks us to show that these are form the vertices of a parallelogram.

00:11 So to do that, we're going to find the vectors that connect these four points.

00:16 We're going to find the vectors ab, ad, cb, and cd.

00:25 And then, so we're going to find these vectors just by subtracting the x, y, and z values.

00:33 From different points, from like point a and point b, the difference between the values of the x, y, and z coordinates will give us the vector ab.

00:45 So once we do that, then if ab is equal to negative cd, so if you multiply one of these by negative one and it gives you the other, that means that it is a parallelogram.

01:01 And same with ad should be equal to negative cb.

01:09 Ok, so once we check that, then we can determine if they form a parallelogram or not.

01:15 And then for the second part, we're asked to find the area of the parallelogram.

01:20 So to do that, we're going to take two vectors that are adjacent.

01:24 So let's say ab and ad.

01:27 We're going to find the cross product of these two vectors.

01:33 And once we get the cross product, then we're going to take the length of that cross product, like the value of that, that will be equal to our area of our parallel.

01:42 Then lastly, we need to determine whether the parallelogram is a rectangle or not.

01:47 To do that, we're going to check the angle between two adjacent points to see if it's a 90 -degree angle or not.

01:53 So the formula that we're going to use for that is sine theta is equal to the length of this cross -product that we just got, which is basically our area.

02:02 It's a -b -b -a -d over value the length of the vector.

02:08 Ab multiplied by the length of the vector ad.

02:13 And it will be a rectangle if theta is equal to 90 degrees.

02:17 All right, so let's test it out then.

02:23 Let's start working on this problem and see what we get.

02:25 So let's start by finding our vectors.

02:30 So let's find vector a, b.

02:33 This is the vector that connects, goes from point a to point b.

02:37 So we're just going to take for our vector.

02:40 We're going to take each value of b and subtract the value on a, and that will give us our vector.

02:47 So for our x value, we've got three for b and two for a, so three minus two is one.

02:53 Then one minus negative one up here is two.

02:58 Then two minus four gives us minus two.

03:03 So this is our vector ab.

03:05 Let's move on to ad.

03:09 So this is the same thing except we're going to subtract.

03:12 Each value in d, take each value in d and subtract the a value.

03:17 We've got minus 1, minus 2, it gives us minus 3, and 3 minus negative 1 gives us 4, and then 8 minus 4 also gives us 4.

03:31 All right, so we have ab and ad.

03:33 Now let's move on to cb and cd.

03:36 So c, b will be equal to, it's going from c to b, so we're going to take each value from b and subtract the corresponding value on c.

03:46 So we've got, um, move this up a little bit.

03:50 We've got three minus zero that gives us three.

03:56 And for the y value, one minus three gives us negative two.

04:01 I'm sorry, taking from b and c.

04:05 So three minus zero gives us three.

04:06 One minus five gives us negative four.

04:09 And two minus six gives us negative four.

04:12 Lastly, let's move on to c, d, which will be equal to, we're going to take each d value and then subtract the corresponding c value.

04:21 So minus 1, minus 0 gives us minus 1.

04:25 3 minus 5 gives us minus 2.

04:29 Then 8 minus 6 gives us 2.

04:34 All right.

04:35 So we've got our four vectors that form our parallelogram, what we think is the parallelogram.

04:40 So now we need to check to make sure it's a parallelogram.

04:43 So remember for it to be a parallelogram, a -b must be equal to minus cd.

04:48 So let's take the minus the c -d.

04:49 So let's take the minus the c -d.

04:50 So multiplying the cd by negative 1, that gives us 1, so we just flip the sign of each component.

04:57 So 1, 2, negative 2.

05:00 As you can see, that's exactly the same as what ab is.

05:03 So we're good here.

05:04 So it means that we have the lines ab and cd are parallel.

05:08 So also, cb should be equal to minus ad.

05:17 So let's see, what's minus ad? so let's take each component of ad and multiply it by negative 1.

05:22 So when we do that, we have three minus four, minus four.

05:27 So we can see that's exactly the same as cb.

05:29 So that means that these two lines are parallel, meaning that our vertices do form a parallelogram.

05:36 So for part a, the answer is yes, they do form a parallelogram.

05:42 All right, so moving on to part b, now we need to find the area of our parallelogram.

05:47 So to do that, we're going to take two adjacent vectors on parallelogram.

05:53 So let's take vectors a, b.

05:55 And ad...

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