Answer:
(A⃗ ×B⃗ )⋅C⃗ = - 76.415
Step-by-step explanation:
First we need to calculate (A⃗ ×B⃗ ) :
(A⃗ ×B⃗ ) = A.B.sin (α).n
Where A is the magnitude of A⃗
Where B is the magnitude of B⃗
Where α is the angle between A⃗ and B⃗ = 63.9 - 25.6 = 38.3
Finally n is the vector orthogonal to A⃗ and B⃗
n magnitude is 1 and his direction is given by the right hand-rule
so n = ( 0 , 0 , 1 )
(A⃗ ×B⃗ ) = A.B.sin (α).n = 5.08 . 3.94 . sin (38.3) . (0 , 0 , 1 ) = (0,0,12.4)
C⃗ can be written as C.(0,0,-1) because of his +z - direction
C.(0,0,-1) = 6.16.(0,0,-1) = (0,0,-6.16)
(A⃗ ×B⃗ )⋅C⃗ = (0,0,12.4).(0,0,-6.16) = -76.41480787 = -76.415
Given plane Π : f(x,y,z) = 4x+3y-z = -1
Need to find point P on Π that is closest to the origin O=(0,0,0).
Solution:
First step: check if O is on the plane Π : f(0,0,0)=0 ≠ -1 => O is not on Π
Next:
We know that the required point must lie on the normal vector <4,3,-1> passing through the origin, i.e.
P=(0,0,0)+k<4,3,-1> = (4k,3k,-k)
For P to lie on plane Π , it must satisfy
4(4k)+3(3k)-(-k)=-1
Solving for k
k=-1/26
=>
Point P is (4k,3k,-k) = (-4/26, -3/26, 1/26) = (-2/13, -3/26, 1/26)
because P is on the normal vector originating from the origin, and it satisfies the equation of plane Π
Answer: P(-2/13, -3/26, 1/26) is the point on Π closest to the origin.
Answer:
area A(w) of the bulletin board as a function of its width, w =[100-w]*w= 100w-
Step-by-step explanation:
- let, the shape of the bulletin board is a rectangle,
- then the perimeter of it = sum of all sides
= 2[length+width] = 2[l+w]
(let l: length, w : width )
100= l+w ( dividing both the sides by 2)
so, l= 100-w
- area = length*width=l*w=[100-w]*w
- therefore,area A(w) of the bulletin board as a function of its width, w =[100-w]*w= 100w-
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