Question

Given in space the points A(4,7,1), B(2,1,3), and c(0,-1,2) The vectors ū = AB , and ✓ = AC a. (9%) Find ū. v , ū x ū , proj, u b. (3%) Find the area of triangle ABC. c. (3 %) Find the parametric equation of line (AB). d. (3 %) Find the distance from point C to the line (AB). e. (3 %) Find the equation of the plane (ABC).

Answer 1

7 AB u 2 A(4,7,1), B(2, 1, 3) , c(0,1,2) u y Ac? un AB. BA 27 + 9 + 3 ² -47-77 _ ^ -21-6] +2 y Ac? 2 - 3 47 - 79 - ya - 4 1 - 7 t a) ū. 627 - 69 +2w).(-47-89 tk) (-2)(-) + (-6)(-8) + (2)(1) 8 +48 +2 ² 58 mxv. ↑ 9 k 7 (-6+16) -;(-2+8) + k (16-24) -] + 212 2 2 -2 - 6 & -4 & 2 107-63-8k • projou (5.2) 2 58 (58), √(-4) + (-8) ²+1? 2 58 9 187

ь деn ДА ВС + | AC XA2 | 3 | х7) 2 4 | 107 -6 -32 | 1. о., 5, Sq. units 2

c) The direction vector along line AB a Joan 2 -21-6] + 2 h let A (14,7,1) be no yo, 2o an-2, 62-6 ,c.2 (from us?) so A & not at Here y yo tbt i na notat 2) 12 4+2t 22 Zotct ya 7-6t 22 lt2t

1-я, 1 2. Eg of plane passing through 3 nom collinear points А( 4, 7, 1) Ч-9) ( 2, 1, 3) 2-2, Х.-и, 91-9, 1-2 (о, -, 1) , -я, Ч-И 1,-21 y-7 9-7 z -) 1-7 - 8 и – 4 2-) . ) и – 4 2-4 26 - 6 1 О 2 3-) 2 - | — 4 — - -7 (-h) (-4 + 10) - (-7) 1-2 +1) +(-1) (16 – 24 ) . o (-4) (to) - -7)(6) + (2-1)(-8) бя - 4 - 6 + 42 - 82 +8 го |ом - у - 8 2 2 - | — lox +6y + 8 2 : /о 3