Calculus please help!

Question

Q2) A and B are the points in space with position vectors a=(2,−1,0) and b=(4,−3,1). 
Find the position vectors of the points P and Q which divide the line segment AB in the ratios 1 : 2 and 2 : 1 respectively. 
a) P: 

b) Q: 

Q3) Let a=(2,3,5), b=(1,−2,4) and c=(1,k,5). 

a) Find a⋅b. 

b) Find the value of k that makes b and c perpendicular. 

c) A triangle is said to be isosceles when two of its edges are the same length. 

(i) Let p=(2,4,−4), q=(5,1,2) and r=(4,3,−1) be the position vectors of the points P,Q and R. Is the triangle PQR isosceles? 
Y/N

(ii) Find the centroid G of the triangle PQR. 

(iii) Is the vector RG perpendicular to PQ? 
Y/N


Q4) Find the equation of the line through P(−2,−1) and Q(3,0) and 
Find the equation of the line through P(2,3) with slope −1


Q5) For f(x)=4x^2 − 5x + 7 find lim(h→0) f(2+h)−f(2) / h 

 

 

Answer 1

(i) Let p=(2,4,−4), q=(5,1,2) and r=(4,3,−1) 

Answer 2

(2).

The point vectors at A and B are a = (2, − 1, 0) and b = (4, − 3, 1) respectively.

(a).

(b).

Answer 3

(3).                                      

(c).

Answer 4

(5). 

The function f(x)=4x^2 − 5x + 7.                                                                       

Answer 5

(4).

The two points are P(−2, −1) and Q(3, 0).

 

The point P(2,3) and slope −1.

Answer 6

(ii)

Let p=(2,4,−4), q=(5,1,2) and r=(4,3,−1)

Calculate the centroid:

G =((x₁+x₂+x₃ )/3,(y_₁+y_₂+y₃)/3,(z_₁+z_₂+z₃)/3)

    = ((2+5+4 )/3,(4+1+3)/3,(-4+2+(-1))/3)

    =(11/3,8/3,-1)

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(iii)

                =-1+1+0

                =0

Therefore, PQ and RG are perpendicular.