If a(x1, y1), b(x2, y2), c(x3, y3), and d(x4, y4) form two line segments, ab and cd , which of these conditions needs to be met to prove that ab is perpendicular to cd?

see attachment for choices

Step-by-step explanation:

We know that,

If two line segments are perpendicular then the product of their slope is equal to -1,

Also, the slope of a line segment having the end points and is,

So, the slope of line segment AB having end points and is,

Similarly, the slope of line segment CD having end points and is,

Hence, by the above property of perpendicular line segments ,

If AB and CD are perpendicular then,

Third option is correct.

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

there is no question content here, but you for the free points! : d