Verify that the points are the vertices of a parallelogram, and find its area. (2,-3,1),(6,5,-1)

step 1 In this question, we are given four points. I've named them A, B, C, and D. We want to show that

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

Key Concepts

Vector Operations

This concept involves using vector addition and subtraction to represent the sides of geometric figures in space. By finding the difference between coordinates, one can determine the direction and magnitude of sides, which is essential for verifying properties like parallelism and equality of sides in shapes such as parallelograms.

Properties of Parallelograms

A key geometric property of parallelograms is that opposite sides are parallel and equal in length, and the diagonals bisect each other. These properties are used to verify that a given set of points forms a parallelogram by, for example, checking if midpoints of the diagonals coincide.

Cross Product for Area Determination

In vector calculus, the area of a parallelogram defined by two vectors can be calculated using the cross product. The magnitude of the cross product of the vectors representing adjacent sides gives the area of the parallelogram, which is a crucial technique when dealing with coordinates in three-dimensional space.

Transcript

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

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

00:06 We want to show that these points form a parallelogram and to find the area of the parallelogram.

00:13 Now, a parallelogram, a, b, c, d, has two pairs of parallel sides, and a, b, the length is equal to cd, and the length ac is equals to bd.

00:30 Now, to find the vector ab, suppose you have two points.

00:34 Points, p and q, and p is x1, y1 and z1, and q is x2, y2, and z2.

00:47 So p q vector will be taking the components of the end point minus the start point.

00:57 So that would be in the x component will be x2 minus x1, y component will be y2 minus y1, and z component will be y2 minus y1, and z 2 minus 7 1.

01:10 Now for area of parallelogram, it will be taking two adjacent vectors, and that will be a, b, crossing with ac, with the direction pointing out or in.

01:25 So it will be a .b.

01:29 Crossing ac.

01:30 Now, for crossing technique, i will show it in the question itself when i'm doing the answers.

01:38 So let's find ac.

01:41 Vector ac will be taking the points c minus a in its component so it will be three minus two minus six minus three and four minus one so that gives us one minus three three all right let's find bd then bd will be taking the point d minus b so it would be 7 minus 6, 2 minus 5, and 2 minus 1.

02:25 And that will give us 1 minus 3, 3.

02:30 So you can see that ac is equals to bd, the vectors itself.

02:36 Not only are they equal, the length is the same...

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