Consider the three points P1(1, 2, 0), P2(3, 6, 20) and P3(2, 4, 6) and a view…
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Consider the three points P1(1, 2, 0), P2(3, 6, 20) and P3(2, 4, 6) and a view point C(0, 0, -10).
Choose the correct options.
(A) P1 obscure P2, if viewed from C.
(B) P2 obscure P1, if viewed from C.
(C) P3 does not obscure P1, if viewed from C.
(D) P2 does not obscure P3, if viwed from C.
Choose the correct answer from the options given below :
- A.
(A), (B) and (C) Only
- B.
(A), (C) and (D) Only
- C.
(B), (C) and (D) Only
- D.
(A), (B) and (D) Only
Attempted by 66 students.
Show answer & explanation
Correct answer: B
Key idea: a point can obscure another from the viewpoint only if both points lie exactly on the same ray from the viewpoint and one lies between the viewpoint and the other.
Compute direction vectors from the viewpoint C(0,0,-10):
C→P1 = P1 − C = (1, 2, 0) − (0, 0, −10) = (1, 2, 10).
C→P2 = (3, 6, 20) − (0, 0, −10) = (3, 6, 30).
C→P3 = (2, 4, 6) − (0, 0, −10) = (2, 4, 16).
Check collinearity and ordering along rays:
C→P2 = 3 × C→P1 (since (3,6,30) = 3·(1,2,10)), so P2 lies on the same ray as P1 but farther from C. Therefore P1 is between C and P2 and will obscure P2 when viewed from C.
C→P3 is not a scalar multiple of C→P1 or C→P2 (the ratios are inconsistent), so P3 is not collinear with P1 or P2 and thus cannot obscure or be obscured by them along the same ray.
Conclusions:
P1 obscures P2 when viewed from C (true).
P2 obscures P1 when viewed from C (false).
P3 does not obscure P1 when viewed from C (true).
P2 does not obscure P3 when viewed from C (true).
Therefore, the true statements are: "P1 obscure P2, if viewed from C.", "P3 does not obscure P1, if viewed from C.", and "P2 does not obscure P3, if viewed from C."
Final answer: the correct combination is the three true statements listed above (P1 obscures P2; P3 does not obscure P1; P2 does not obscure P3).