ABCDEF a = AB, b = a b (1) AC (3) CD (2) AD (4) CE AF B C a A D b F E (1) AC = AB + BC = AB + AO = AB + ( AB + AF) = a + ( a + b) = 2 a + b (2) AD = 2 AO = 2( AB + AF) = 2( a + b) (3) CD = AF = b (4) CE = CO + OE = BA + AF = AB + AF = a + b B C a A b O D F E [1] ABCD O AB 1 : 2 E, BC F AB, AD (1) AO (2) DO (3) AF (4) EF [2] x, y a, b (1) x + 2( x b) = 4( b 2 a) a (2) 1 2 ( b + x) + 3( x 1 2 b) = 0 (3) 2 x + 3 y = a + b, 3 x + 2 y = b 1
2 (1) a = ( 2, 2), b = (1, 2), c = (4, 4) c = l a + k b l, k (2) a = (3, 5) (1) (4, 4) = l( 2, 2) + k(1, 2), (4, 4) = ( 2l + k, 2l 2k) 2l + k = 4, 2l 2k = 4 l = 6, k = 8 (2) 1 a 1 1 a = (3, 5) = (3, 5) 32 + 52 34 [3] a = ( 1, 4), b = (3, 2), c = (0, 5) (1) 3 a + b (2) a 2 b + 3 c (3) 6( c 2 a) 5( 3 b + a) [4] (1) a = (1, 1), b = ( 1, 3), c = ( 3, 5) c = l a + k b l, k (2) ABCD A(1, 2), B(3, 5), C( 4, 0) D [5] a = (3, 1), b = (1, 2), c = a + t b t (1) c = 15 t (2) c t [6] A, B, C AB = a, AC = b ABC a, b 2
3 (1) a = (x 1, y 1 ), b = (x 2, y 2 ), a b θ a b cos θ = x 1 x 2 + y 1 y 2 (2) a = (1, 1), b = (2, x) x (1) a b (2) a b 120 (3) a b (1) O(0, 0), A(x 1, y 1 ), B(x 2, y 2 ) AB 2 = OA 2 + OB 2 2OA OB cos θ (x 2 x 1) 2 + (y 2 y 1) 2 = (x 2 1 + y1) 2 + (x 2 2 + y2) 2 2 a b cos θ a b cos θ = x 1x 2 + y 1y 2 (( )) a b = a b cos θ = x 1 x 2 + y 1 y 2 (2) (1) a b = 0 1 2 + ( 1) x = 0 x = 2 (2) a b = a b cos 120 1 2 + ( 1) x = p 1 2 + ( 1) 2 2 2 + x 2 ( 1 2 2 ), 2 x = 4 + x 2 2 ( 2(2 x)) 2 = ( 4 + x 2 ) 2, 2(4 4x + x 2 ) = 4 + x 2, x 2 8x + 4 = 0, x = 4 2 3 x = 4 + 2 3 (3) b = k a (2, x) = k(1, 1), 2 = k x = k x = 2 [7] a = 4, b = 5 a b a b (1) 45 (2) 120 (3) 90 (4) 180 [8] a, b (1) a = (2, 0), b = (2, 1) (2) a = (1, 1), b = (3, 2) (3) a = (k + 2, k 1), b = (2k 4, k + 1) (4) a = (p + q, q), b = (p q, p) [9] a, b (1) a = (1, 3), b = (2, 0) (2) a = (3, 7), b = (2, 5) (3) a = ( 3 1, 3 1), b = (1, 1) (4) a = ( 2, 2), b = ( 3 1, 3 + 1) [10] (1) a = 1, b = 5, 2 a + b = 3 a b (2) a = (3, 7), b = (2, 5) a + t b t 3
4 ABC AB) 3 : 1 D AC 4 : 3 E BE CD P AP BC Q (1) BE, CD AB, AC (2) AP : P Q [11] OAB OA 5 : 2 C OB 3 : 4 D CD M (1) OM OA, OB (2) OM AB N ON : OM, AN : NB [12] ABCDE AB = a, BC = b CD a, b [13] ABC AB = c, BC = a, CA = a AB = c, BC = a, CA = b (1) ABC G AG b c (2) AG 2 a, b, c (3) a = 12, b = 21, c = 3 AGB 4
5 [1] (1) ( 3, 4) v = (1, 2) (2) (1, 3), ( 2, 4) (3) (5, 4) v = (1, 2) [2] (2, 0) ( 1, 3) [14] (1) (, 4) n = (1, 2) (2) (1, 3), ( 2, 4) [15] ( 2, 3) x y 1 = 0 [16] ABC O OA = a, OB = b, OC = c (1) A BC A AA (2) ABC H OH a, b, c 5
6 OABC AB 1 : 2 D CD 3 : 5 E OE 1 : 3 F AF OBC G OA = a, OB = b, OC = c (1) OE a, b, c (2) AG : F G [17] ABCD a, b, c, d 3 : 1 [18] ABCD 3 A(1, 2, 3), B(3, 2, 1), C(6, 4, 4) (1) D (2) E(1, y, 15) ABC y [19] P ABC A P BC H P A = a, P B = b, P C = c (1) a b, b c, c a (2) P H b, c (3) P ABC 6
7 [1] (1) (1, 2, 3) v = (5, 2, 7) (2) (1, 1, 1), ( 2, 1, 3) [2] (1) (1, 2, 3) v = (5, 2, 7) (2) 3 (2, 1, 1), (3, 1, 1), (4, 1, 1) [20] (1) (1, 2, 3) v = (2, 3, 1) (2) ( 1, 5, 2), (3, 4, 1) [21] (1) (4, 2, 1) v = (1, 1, 3) (2) 3 ( 2, 3, 1 ), (2, 2, 3), ( 4, 1, 1) 7
* 8 [1] x 1 3 = y + 1 2 [2] = z + 2, 3x + 2y + z = 1 2 (1, 1, 2), 2x y + z 1 = 0 [22] (1) x + 2 2 = y 3 = y 1, 2x y + 3z + 2 = 0 3 (2) x + 1 2 = y 1 3 [23] (1) (2, 0, 1), x + y 2z + 3 = 0 = z + 2, 2x 3y + z 1 = 0 2 (2) ( 1, 1, 1), 3x y + 2z + 1 = 0 [24] (1, 2, 3) (1) 2x + 3y + z + 1 = 0 (2) x + 2y 2z 3 = 0 8
* 9 (1) a b = b a (2) a ( b + c) = ( a b) + ( a c) (3) (k a) b = k( a b) = a (k b) (4) ( a b) c = ( a c) b ( b c) a (5) ( a b) a ( a b) b (6) ( a b) 2 = ( a a)( b b) ( a b) 2 [25] a b a b (1) a = (1, 1, 1), b = ( 2, 3, 1) (2) a = ( 1, 1, 2), b = (1, 0, 1) [26] a = (1, 1, 3), b = ( 3, 1, 4) [27] a = (1, 0, 2), b = ( 2, k, 4) a b = 0 k [28] 3 a, b, c c ( a b) 9
[1] OABC OAB P, OBC Q OA = a, OB = b, OC = c PQ a, b, c PQ = x a + y b + z c z = AOC = 60, OA = 3, OC = 2 PQ = ( ) [2] OAB OA = 2, OB = 3, AOB = 120 AB M, AM N OA = a, OB = b (1) a b (2) OM, ON a, b (3) OM ON (4) MON = θ cos θ ( ) [3] 2 a = (1, x), b = (2, 1) (1) a + b 2 a 3 b x (2) a + b 2 a 3 b x ( ) [4] ABCD AB // DC, AB = 6, CD = 4 AB, AD M, N MN P MP : PN = 1 : 3 CP AB Q AB = a, AD = b (1) AC, AP a, b (2) CP : PQ (3) AD = 5, BAD = 60 CQ ( ) 10
[5] OABC OA = 3, OB = 4, OC = 5 AOB, AOC, BOC 60 BC O BC OP OB = b, OC = c OP = b + c OP = A OBC AQ OQ = b + c AQ = OABC ( ) [6] 3 a = (cos α, sin α, 0), b = (sin α, cos α, t), c = (sin α, cos α, 0) α, t a, b 0 v v c θ cos θ ( ) [7] (0, 0, 0) O A(1, 0, 0), B(2, 1, 0), C(3, 4, 1) OA = a, OB = b, OC = c 3 A, B, C α (a) P(x, y, z) OP = r a + s b + t c r, s, t x, y, z (b) P α x, y, z (c) (4, 5, 7) D α H DH AB, DH BC H ( ) [8] 3 O(0, 0, 0), A(1, 1, 0), B(1, 0, 1) α (1) α OA (2) α OAB ( ) 11