Shifting the Origin:
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Shifting origin to another point without changing directions of the axes is called as translation of axes.
Let origin O be shifted to another point
O' (x', y', z') without changing direction of axes. Let the new coordinate frame be O'X'Y'Z'. Let P (x, y, z) be a point with respect to coordinate frame OXYZ.
Then, coordinate of point P with respect to new coordinate frame O'X'Y'Z' is (x1, y1, z1), where
x1 = x - x', y1 = y - y', z1 = z - z'
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Example: If origin is shifted (1, 2, -3) without changing the directions of the axes then find the new coordinates of point (0, 4, 5) with respect to new frame.
Solution: x' = x - x1, here (x1, y1, z1) is the shifted origin
y' = y - y1
z' = z - z1
x' = 0 - 1 = -1
y' = 4 - 2 = 2
z' = 5 + 3 = 8
∴ The coordinates of the point with respect to new coordinate frame is (-1, 2, 8).
Note:
- Distance between points P(x1, y1, z1) and Q (x2, y2, z2) is
- The point dividing line joining P(x1, y1, z1) and Q(x2, y2, z2) in m : n ratio is
where m + n ≠ 0 .
- The coordinates of centroid of the triangle having vertices A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is G
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