use a graph paper and form the quadrilateral defined by the given vectors...the rest is easy...
I know I have to show that two sides are equal and in order to prove this is a parallelogram, however I am not sure which sides to pick ?
i.e.
Is it PQ = RS
or
RP = SQ,
How would I know, and I dont think i would be able to sketch the diagram as its in a R^{3}
I find working with graph paper in difficult. The four points may not even be co-planar .
The four points determine six vectors. In order to have a parallelogram some pair of those six vectors must be parallel and have the same length.
Recall that parallel vectors are multiples of each other.
no.
First make a rough sketch so you can tell which points are which vertices.
Then you can determine the vector of each of the four sides as the differences of the position vectors of the appropriate vertices.
To show the opposite sides are parallel you can show their cross products are zero.
Then use the cross product on the vectors of two adjacent sides to find the area.
Without any pre-conceptions of where P,Q,R,S are:
1) Find PQ,PR,PS: two of these have to be adjacent sides of a quadrilateral.
2) Find QR,QS: One of these has to be a side other than the two adjacent sides of 1)
If one of 1) is equal to one of 2) it’s a parallelogram.
Sounds good, but really not intelligible (to me) without playing with some sketches of four arbitrary points.
Hello, Tweety!
Given points: .
Show that is a parallelgram and use the cross-product to find its area.
I assumed that PQRS are in order around the parallelogram.
Then I checked this assumption.
Code:(-1,0,2) (3,4,-1) P *---------* Q / / / / / / S *---------* R (-1,2,0) (3,2,-3)
If two sides of a quadrilateral are parallel and equal,
, , the quadrilateral is a parallelogram.
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