Verify that the points are the vertices of a parallelogram, and find its area. (1,1,1),(2,3,4),(

step 1 In this question, we are given four points. I've named them A, B, C and D. We are to show that t

Verify that the points are the vertices of a parallelogram,...

Key Concepts

Area Calculation Using the Cross Product

Once a quadrilateral is verified to be a parallelogram, its area in three dimensions can be computed by taking the cross product of two adjacent side vectors. The magnitude of the cross product gives the area of the parallelogram, making this method both effective and straightforward in multidimensional settings.

Parallelogram Properties in Coordinate Geometry

This concept involves understanding the criteria for a quadrilateral to be a parallelogram. In coordinate geometry, a common method is to show that either both pairs of opposite sides are parallel and equal in length, or that the diagonals bisect each other. These properties ensure that the shape has the symmetry and balance characteristic of a parallelogram.

Vector Operations in Three Dimensions

Utilizing vectors to analyze geometric figures in three dimensions is crucial. By representing the sides of the quadrilateral as vectors, one can easily test for equality and parallelism by comparing the vector components. This approach is essential when verifying the necessary properties to classify a shape as a parallelogram.

Transcript

00:01 In this question, we are given four points.

00:03 I've named them a, b, c and d.

00:05 We are to show that these four points form a parallelogram and to find its area.

00:11 Now, for a parallelogram, ab will be parallel to cd and ac will be parallel to bd.

00:24 And not only that, the length of ab will be same as cd and the length of ac will be same as bd.

00:31 Now to find your vectors, for instance, between two points, p and q, and q having the point x1, y1, z1, q having the point x2, y2, z2, the vector pq will be taking the component of the end point minus the start point.

00:56 So that will be x2 minus x1, y2 minus y1, and z2 minus z1.

01:11 For area of parallelogram, in this case, abcd, will be the magnitude of taking the two adjacent vectors going out or in.

01:23 So i'll be taking ab crossing with ac.

01:29 For the crossing technique, i will show you in the solution itself.

01:35 Okay, so now let's find factors ab.

01:41 We'll be taking point b minus point a in its component form.

01:45 So it will be 2 minus 1, 3 minus 1, and 4 minus 1.

01:54 So that gives us 1, 2, 3.

01:59 Let's look at, so i've done ab, so let's look at cd.

02:04 Factor cd will be taking d minus c, 7 minus 6, 7 minus 5, and 5 minus 2.

02:18 So this gives us 1, 2 3 as well.

02:21 So you can see that ab factor is as a.

02:25 Same as cd vector so not only you're shown that they are of parallel same direction you're also shown they are equal length so we're done ab and cd so let's look at ac now ac is this one over here so ac taking c minus a so it'll be six minus one five minus one and two minus one and that gives us 5 for 1...

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