Answer to Question #200627 in Analytic Geometry for Vina

Question

Let P = (1, 2, 3) and Q = (−5, −1, 12).

(4.1) Find the midpoint of the line segment connecting P and Q.

(4.2) Find the point on the line segment connecting P and Q that is 2/3 of the way from P to Q.

(4.3) Let P = (−1, 5, 2). If the point (0, −2, 3) is the midpoint of the line segment connecting P and Q, what is the point Q?

Answer 1

$P(1,2,3), Q(-5,-1,12)$

4.1

$x_A =\frac{x_P +x_Q}{2}=\frac{1-5}{2}=-2\\ y_A =\frac{y_P +y_Q}{2}=\frac{2-1}{2}=0.5\\ z_A =\frac{z_P +z_Q}{2}=\frac{3+12}{2}=7.5\\$

A(-2, 0.5, 7.5) the midpoint of the line segment connecting P and Q.

4.2

$x_B=\frac{x_P+\lambda x_Q}{1+\lambda}=\frac{1+\frac{2}{3}\cdot(-5)}{1+\frac{2}{3}}=-\frac{7}{5}\\ y_B=\frac{y_P+\lambda y_Q}{1+\lambda}=\frac{2+\frac{2}{3}\cdot(-1)}{1+\frac{2}{3}}=\frac{4}{5}\\ z_B=\frac{z_P+\lambda z_Q}{1+\lambda}=\frac{3+\frac{2}{3}\cdot12}{1+\frac{2}{3}}=\frac{33}{5}\\$

B($\frac{7}{5},\frac{4}{5},\frac{33}{5}$ ) the point on the line segment connecting P and Q that is $\frac{2}{3}$ of the way from P to Q.

4.3

$P(-1,5,2) , O(0,-2,3)$

O is the midpoint of the line segment connecting P and Q

$Q(x,y,z)\\ x_O=\frac{x_P+x_Q}{2}\\ 0=\frac{-1+x}{2}\\ x=1\\ y_O=\frac{y_P+y_Q}{2}\\ -2=\frac{5+y}{2}\\ y=-9\\ z_O=\frac{z_P+z_Q}{2}\\ 3=\frac{2+z}{2}\\ z=4$

Q(1,-9,4)