Math Problem Statement

Line segment is drawn between points A(2,-4,3) and B(6,8,-5). A. Find point that divides the segment in the ratio 4:3. B,if point P is shifted by (1,-2,3) determine new co ordinates of P

Solution

Part A: Find the point that divides the segment in the ratio 4:3

The formula to find the coordinates of the point P(x,y,z)P(x, y, z) dividing a line segment joining points A(x1,y1,z1)A(x_1, y_1, z_1) and B(x2,y2,z2)B(x_2, y_2, z_2) in the ratio m:nm:n is:

P(x,y,z)=(mx2+nx1m+n,my2+ny1m+n,mz2+nz1m+n)P(x, y, z) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n} \right)

Here, A(2,4,3)A(2, -4, 3), B(6,8,5)B(6, 8, -5), m=4m = 4, and n=3n = 3.

Calculation of P(x,y,z)P(x, y, z):

x=4(6)+3(2)4+3=24+67=307x = \frac{4(6) + 3(2)}{4+3} = \frac{24 + 6}{7} = \frac{30}{7}

y=4(8)+3(4)4+3=32127=207y = \frac{4(8) + 3(-4)}{4+3} = \frac{32 - 12}{7} = \frac{20}{7}

z=4(5)+3(3)4+3=20+97=117z = \frac{4(-5) + 3(3)}{4+3} = \frac{-20 + 9}{7} = \frac{-11}{7}

So, the point dividing the segment in the ratio 4:3 is:

P=(307,207,117)P = \left( \frac{30}{7}, \frac{20}{7}, \frac{-11}{7} \right)


Part B: Shift PP by (1,2,3)(1, -2, 3)

If P(x,y,z)P(x, y, z) is shifted by (1,2,3)(1, -2, 3), the new coordinates P(x,y,z)P'(x', y', z') are calculated as:

x=x+1,y=y2,z=z+3x' = x + 1, \quad y' = y - 2, \quad z' = z + 3

Calculation of new coordinates:

x=307+1=307+77=377x' = \frac{30}{7} + 1 = \frac{30}{7} + \frac{7}{7} = \frac{37}{7}

y=2072=207147=67y' = \frac{20}{7} - 2 = \frac{20}{7} - \frac{14}{7} = \frac{6}{7}

z=117+3=117+217=107z' = \frac{-11}{7} + 3 = \frac{-11}{7} + \frac{21}{7} = \frac{10}{7}

The new coordinates of PP are:

P=(377,67,107)P' = \left( \frac{37}{7}, \frac{6}{7}, \frac{10}{7} \right)


Final Answer:

A. The point dividing the segment in the ratio 4:34:3 is:

P=(307,207,117)P = \left( \frac{30}{7}, \frac{20}{7}, \frac{-11}{7} \right)

B. After shifting PP by (1,2,3)(1, -2, 3), the new coordinates are:

P=(377,67,107)P' = \left( \frac{37}{7}, \frac{6}{7}, \frac{10}{7} \right)


Do you want a step-by-step breakdown of any part?

Expanding Questions:

  1. How do you determine the ratio when dividing a segment between points in space?
  2. What happens if the segment is divided externally instead of internally?
  3. How can the same formula be extended to higher dimensions (e.g., 4D or 5D)?
  4. What is the significance of the shift vector in coordinate transformations?
  5. How do different ratios affect the position of the dividing point?

Tip:

Always simplify the ratio into its lowest terms before applying the section formula to avoid unnecessary

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Ratios
Vector Shifts

Formulas

Section formula: P(x, y, z) = [(mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n)]
Vector shift: P'(x', y', z') = (x + dx, y + dy, z + dz)

Theorems

Section formula theorem for dividing a line segment in a given ratio

Suitable Grade Level

Grades 9-11