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556636.com:高中数学(人教A版,必修4)分层测评:2.4.2 平面向量数量积的坐标表示、模、夹角

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学业分层测评(十九) (建议用时:45 分钟) [学业达标] 一、选择题 1.已知向量 a=(3,1),b=(x,-2),c=(0,2),若 a⊥(b-c),则实数 x 的值为( 4 A. 3 3 C.- 4 3 B. 4 4 D.- 3 ) 4 【解析】 b-c=(x,-4),由 a⊥(b-c)知 3x-4=0,∴x= .故选 A. 3 【答案】 A 2.已知向量 a=(1,-2),b=(x,4),且 a∥b,则|a-b|=( A.5 3 C.2 5 B.3 5 D.2 2 ) 【解析】 ∵a∥b,∴4+2x=0, ∴x=-2,a-b=(1,-2)-(-2,4)=(3,-6), ∴|a-b|=3 5.故选 B. 【答案】 B 3.已知向量 a=(1, 3),b=(-2,2 3),则 a 与 b 的夹角是( π A. 6 π C. 3 π B. 4 π D. 2 ) 【解析】 设 a 与 b 的夹角为 θ, , 3? ·? -2,2 3? 1 a· b ?1 则 cos θ= = = , |a||b| 2× 4 2 π 解得 θ= .故选 C. 3 【答案】 C 4.若 a=(2,3),b=(-4,7),则 a 在 b 方向上的投影为( A. C. 65 5 13 5 B. 65 D. 13 a 在 b 方向上的投影为 |a|cos 〈 a , b 〉= ?-4,7? a· b ?2,3?· = = |b| ?-4?2+72 ) 【导学号:00680059】 【解析】 2× ?-4?+3× 7 65 = . 5 65 【答案】 A 5.已知正方形 OABC 两边 AB,BC 的中点分别为 D 和 E,则∠DOE 的余弦值为( 1 A. 2 3 C. 5 B. 3 2 ) 4 D. 5 1 1, ? , 【解析】 以点 O 为原点, OA 所在直线为 x 轴建立直角坐标系, 设边长为 1, 则 D? ? 2? 1 ? E? ?2,1?,于是 cos∠DOE= 【答案】 D 二、填空题 → → → → → → 6.已知 O A =(-2,1),O B =(0,2),且 A C ∥O B ,B C ⊥A B ,则点 C 的坐标是____ . → 【解析】 设 C(x,y),则 A C =(x+2,y-1), → → B C =(x,y-2),A B =(2,1). → → → → 由 A C ∥O B ,B C ⊥A B ,得 ?2?x+2?=0, ?x=-2, ? ? ? 解得? ? ? ?2x+y-2=0, ?y=6, ?1,1?· ?1,1? ? 2? ?2 ? 4 = . 1 1 ?2 ? ?2 2 5 12+? ?2? · ?2? +1 ∴点 C 的坐标为(-2,6). 【答案】 (-2,6) 7.若向量 a=(-2,2)与 b=(1,y)的夹角为钝角,则 y 的取值范围为________. 【解析】 若 a 与 b 夹角为 180° ,则有 b=λa(λ<0), 1=-2λ, ? ? 即?y=2λ, ? ?λ<0, 1 解得 y=-1 且 λ=- ,所以 b≠λa(λ<0)时,y≠-1;① 2 π ? 若 a 与 b 夹角 θ∈? b<0 且 b≠λa(λ<0). ?2,π?时,则只要 a· 当 a· b<0,有-2+2y<0,解得 y<1.② 由①②得 y<-1 或-1<y<1. 【答案】 (-∞,-1)∪(-1,1) 三、解答题 → → → 8.已知AB=(6,1),BC=(4,k),CD=(2,1). (1)若 A,C,D 三点共线,求 k 的值; → → (2)在(1)的条件下,求向量BC与CD的夹角的余弦值. 【导学号:70512037】 → → → 【解】 (1)因为AC=AB+BC=(10,k+1),由题意知 A,C,D 三点共线, → → 所以AC∥CD,所以 10× 1-2(k+1)=0,即 k=4. → → BC· CD 12 3 10 → → → (2)因为CD=(2,1),设向量BC与CD的夹角为 θ,则 cos θ= = = . → → 4 2× 5 10 |BC||CD| 9.已知 a=(1,1),b=(0,-2),当 k 为何值时, (1)ka-b 与 a+b 共线; (2)ka-b 与 a+b 的夹角为 120° . 【解】 ∵a=(1,1),b=(0,-2), ka-b=k(1,1)-(0,-2)=(k,k+2), a+b=(1,1)+(0,-2)=(1,-1). (1)∵ka-b 与 a+b 共线, ∴k+2-(-k)=0,∴k=-1. 即当 k=-1 时,ka-b 与 a+b 共线. (2)∵|ka-b|= k2+?k+2?2, |a+b|= 12+?-1?2= 2, (ka-b)· (a+b)=(k,k+2)· (1,-1) =k-k-2=-2, 而 ka-b 与 a+b 的夹角为 120° , ∴cos 120° = ?ka-b? ·? a+b? , |ka-b||a+b| -2 1 即- = , 2 2 2· k +?k+2?2 化简整理,得 k2+2k-2=0,解之得 k=-1± 3. 即当 k=-1± 3时,ka-b 与 a+b 的夹角为 120° . [能力提升] 1. 已知向量 a=(1,2), b=(2, -3). 若向量 c 满足(c+a)∥b, c⊥(a+b), 则 c 等于( 7 7? A.? ?9,3? 7 7? C.? ?3,9? 7 7? B.? ?-3,-9? 7 7? D.? ?-9,-3? ) 【解析】 设 c=(x,y), 又因为 a=(1,2),b=(2,-3), 所以 c+a=(x+1,y+2). 又因为(c+a)∥b, 所以有(x+1)· (-3)-2· (y+2)=0, 即-3x-2y-7=0,① 又 a+b=(3,-1), 由 c⊥(a+b),得 3x-y=0,② ?x=-9, 由①

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