Verify that the points are the vertices of a parallelogram and find its area: (2,-1,1),(5,1,4),(

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

Key Concepts

Area Calculation

Area calculation of a parallelogram in three dimensions involves finding the magnitude of the cross product of two adjacent side vectors. This method provides a reliable means to determine the area of parallelograms using vector operations.

Cross Product

The cross product is an operation in three-dimensional vector algebra that produces a new vector perpendicular to the plane formed by the original two vectors. Its magnitude equals the area of the parallelogram defined by those vectors, making it an essential tool for calculating area in 3D geometry.

Parallelogram Properties

Parallelogram properties refer to the geometric characteristics that define a parallelogram, including that opposite sides are equal in length and parallel. This concept is crucial for verifying that a set of points forms a parallelogram by checking that corresponding sides have identical vector representations.

Vectors

Vectors represent directed line segments in space, which can be used to describe the sides of geometric figures. By constructing vectors from the vertices, one can analyze the relationships between sides, such as parallelism and equality in length.

Transcript

00:01 Hello students here we have a question that we have to verify the points are vertices of parallelogram and we have to find its area let's we have a parallelogram of point a b c and d so if these points are what this is or parallelogram then vertex then distance from a to b should be equals to cd and a to c have to equals to bd and the diagonal c d c b b not equals to a d.

01:06 If these three points are verified or if these three points are valid for this given data then we can say that these points are the vertices of a parallelogram so first of all we will find a b a b will be equals to point a is 2 minus 1 and 1 minus point b is 5 1 and 4 so it will be equals to 5 minus 2 is 3 comma of 1 minus minus 1 will be equal to 2 and 4 minus 1 would be equals to 3 these magnitudes are equal and these two vectors magnitude should have not be equal so the magnitude of ab will be equals to under root three square plus two square plus three square which is equals to nine plus nine plus four is equals to twenty two now we'll find cd cd is equals to c is 011 and d is 3 334 011 minus 3 3 4 so it will be equal to 3 minus 0 0 minus 1 is 2 .4 minus 1 is 3 so the magnitude of cd will be equals to 3 square plus 2 square plus 3 square which is equal to under root 22 so here we proved that a b magnitude is equals to the magnitude of cd this line and this line is equal now we will find ac and b d so ac will be equals to vector a is 2 minus 1 and 1 2 minus 1 and 1 and the point c is 011 011 1 1 so 0 minus 2 is minus 2 1 minus minus 1 is 2 and 1 minus 1 will be equal to 0 now the magnitude of ac will be equals to 2 2 2 square plus 2 square plus 0 which is equal to under root 8 then we will find bd bd will be equals to bd is 514 5 1 and 4 minus vector d is 3 3 3 4 3 4 that will be goes to 3 minus 5 is minus 2 and 4 is 0 the magnitude of 8 will be equals to minus two square plus two square plus zero square which is equals to eight so the magnitude of ac is equals to magnitude of bd so here this line and this line is also equal now we have to show that the diagonals are not equal if the diagonals are equal then it is a triangle and if the diagonals are not equal then we can say that it's a perpendicular okay the diagonals are cb and a d first of all we will find cd cd will be equals to c is 011 minus d is 3 334 here we can see that c is 011 and d is 3 334 cd will be equals to 3 minus 0 is 3 comma 3 minus 1 is 2 comma 3 minus 3 now the magnitude of cd will be equals to under root 3 square plus 2 square plus 3 square which is equals to 9 plus 9 is 18 18 plus 4 is 22 under root 22 now we will find the diagonal so ad ad will be equals to is 2 minus 1 1 and 334 sorry here we have to find cd c b we have to find the diagonal cb so c is 011 and b is 514 here we have to find the diagonal c b and b is 514 here we have to find the diagonal c b and b b is 514 so 5 minus 0 is 5 comma 1 minus 0 comma 4 minus 1 is 3 so the magnitude of cd will be equals to 5 square plus 0 square plus 3 square root which is equals to under root 34 now we will find the diagonal a d is 2 1 1 d is 3 3 4 diagnose ad is equals to a is 2 minus 1 1 minus d is 3 3 3 is 1 so 3 minus 2 is 1 b equals to 4 and 3 minus 1 is 3 the magnitude of ad b equals to 1 and the root 1 square plus 4 plus 1 square plus three square which is equals to 16 plus 9 plus 1 17 26 and the root 26 so we can see that the diagonals are not equal because magnitude of bc is not equals to magnitude of ad so we can see that that it is a parallelogram.

09:57 Now we have to find the area...

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