Vector, Vertices (1 Viewer)

[prime-factor]

Show that the following are vertices of a rectangle:

i+j+k (3 dimensions)

A(2,1,3)
B(4,2,-1)
C(5,4,0)
D(3,3,4)

A, B, C and D are position vectors


Apparently since these are just points I have to find a resultant vector(s) and then do some other calculations?

Please help. Just started this topic today.

 

[Iruka]

The diagonals of a rectangle are of equal length and bisect each other.

 

[roadrage75]

Show that the following are vertices of a rectangle:

i+j+k (3 dimensions)

A(2,1,3)
B(4,2,-1)
C(5,4,0)
D(3,3,4)

A, B, C and D are position vectors


Apparently since these are just points I have to find a resultant vector(s) and then do some other calculations?

Please help. Just started this topic today.

have you guys learnt about vectors yet??

if so, you can use the dot product rules.

vector AB = 2i + j -4k
vector CD = -2i -j + 4k = -(2i + j -4k)

now two vectors are parallel if one is the multiple of the other, and thus AB is parallel to CD. (note |AB| = |CD| )

vector AD = i + 2j + k
vector BC = i + 2j + k

and for the same reason, AD is parallel to BC. (note |AD| = |BC| )

so we now know, at the very least, ABCD is a parallelogram, as opposite sides are parallel and equal

To prove that it is a rectangle, we must prove that one of the angles is 90 degrees.

if AB and AD are perpendicular, then AB . BC = 0 (the dot product)


so what does (2i + j - 4k) . (i + 2j + k) = ? (again dot product)

it = 2 + 2 -4 = 0

hence AB and AD are perpendicular.

Since opposite sides of the quadraletal are parallel and equal, and one angle is a right angle, it follows that ABCD is a rectangle.

 

[prime-factor]

Okay. Thanks. We have done vectors at an introductory level, but are now doing alot more work on dot product and cross product.

vector AB = 2i + j -4k

So you did B-A

vector CD = -2i -j + 4k = -(2i + j -4k)

And here you did D-C

What is the name of this rule and where can I learn more about it online?

 

[roadrage75]

Okay. Thanks. We have done vectors at an introductory level, but are now doing alot more work on dot product and cross product.

vector AB = 2i + j -4k

So you did B-A

vector CD = -2i -j + 4k = -(2i + j -4k)

And here you did D-C

What is the name of this rule and where can I learn more about it online?

yeah, that's right, as i think you mentioned, its called the "resultant vector", i guess look up "vectors" and you might find some information...

or look up "resultant vector", "cross product rule", "dot product rule"... this should give you some good information i think