Thread: Two vector questions (high school level)

1) Expand and simplify: (-2a + 3b) dot (5a - b). (a and b being vectors)

2) Explain how to prove if 4 given points are coplanar. Determine if A(3, 1, 0), B(2, -3, 1), C(-1, 0, 4) and D(5, -6, -2) are coplanar.

Thanks, I'm completely stumped on both.

Hi,
a.)

(-2a+3b).(5a-b)
=(-2)(5)+(3)(-1)
=-10-3
=-13

b.)

Give point A(3,1,0); B(2,-3,1): C(-1,0,4); D(5,-6,-2)

A=3i+j; B=2i-3j+k; C=-i+4k; D=5i-6j-2k

5i-6j-2k=x(3i+j)+y(2i-3j+k)+z(-i+4k)
5i-6j-2k=(3x+2y-z)i+(x-3y)j+(y+4z)k
here equation the like vectors
3x+2y-z=5
x-3y=-6
y+4z=-2

solve x , y and z values
and substitute in one of the equation
if its satisfies then the given vectors are coplanar

all the best

I got a question too
for second question can we just do

AB, BC, CD

and then use the triple scalar product
AB . (BC X CD)

if that equals zero then it means they are coplanar

will that work?

Thanks in advance







-----------------------------------
AB = [-1, -4, 1]
BC = [-3, 3, 3]
CD = [6, -6, -6]


AB . (BC X CD)

[-1, -4, 1] . [0, 0, 0]

= 0

which means they are co planar

can anyone plz verify this one from me (whether this method is correct or not)