In the figure below, AC and BE are diameters of circle P. What is the arc measure of DE in degrees?

arc DE = 39°

Step-by-step explanation:

Since BE is a diameter then

∠ BPA + ∠ EPA = 180° ( straight angle ), substitute values

4w + 8 + 4w + 4 = 180, that is

8w + 12 = 180 ( subtract 12 from both sides )

8w = 168 ( divide both sides by 8 )

w = 21

Since AC is a diameter, then

∠ EPA + ∠ EPD + ∠ DPC = 180° ( straight angle ), substitute values

4w + 4 + ∠ EPD + 2w + 11 = 180, that is

6w + 15 + ∠ EPD = 180, substitute w = 21

6(21) + 15 + ∠ EPD = 180

126 + 15 + ∠ EPD = 180

141 + ∠ EPD = 180 (subtract 141 from both sides )

∠ EPD = 39°

The angle at the centre of the circle is equal to the arc subtending it, that is

arc DE = ∠ EPD = 39°

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