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7se8se:2019届高中数学人教A必修4学业分层测评19 平面向量数量积的坐标表示、模、夹角 含解析

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2019 届数学人教版精品资料 学业分层测评(十九) (建议用时:45 分钟) [学业达标] 一、选择题 1.(2016· 开封质检)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 【解析】 ) 3 B.4 4 D.-3 b-c=(x,-4),由 a⊥(b-c)知 3x-4=0, 4 ∴x=3.故选 A. 【答案】 A 2.(2016· 马鞍山质检)已知向量 a=(1,-2),b=(x,4),且 a∥b,则|a-b| =( ) A.5 3 C.2 5 【解析】 ∵a∥b,∴4+2x=0, B.3 5 D.2 2 ∴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, a·b (1, 3)· (-2,2 3) 1 则 cos θ =|a||b|= =2, 2×4 π 解得 θ= 3 .故选 C. 【答案】 C ) 4.若 a=(2,3),b=(-4,7),则 a 在 b 方向上的投影为( 【导学号:00680059】 65 A. 5 13 C. 5 B. 65 D. 13 (-4,7) a· b (2,3)· 【解析】 a 在 b 方向上的投影为|a|cos<a,b>= |b| = (-4)2+72 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 ) 3 B. 2 4 D.5 【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??1 ? ? ?1,2??2,1? ? ?? ? 4 =5. ?1?2 ?1?2 12+?2? · ?2? +12 ? ? ? ? 1? ? ?1 ? 1,则 D?1,2?,E?2,1?,于是 cos∠DOE= ? ? ? ? 【答案】 二、填空题 D → =(-2,1),OB → =(0,2),且AC → ∥OB → ,BC → ⊥AB → ,则点 C 的坐标 6.已知OA 是________. 【解析】 → =(x+2,y-1), 设 C(x,y),则AC → =(x,y-2),AB → =(2,1). BC → ∥OB → ,BC → ⊥AB → ,得 由AC ? ? ?2(x+2)=0, ?x=-2, ? 解得? ? ? ?2x+y-2=0, ?y=6, ∴点 C 的坐标为(-2,6). 【答案】 (-2,6) 7.(2016· 德州高一检测)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 ? ? 1 即?y=2λ ,解得 y=-1 且 λ=-2,所以 b≠λa(λ<0)时 y≠-1;① ? ?λ<0 ?π ? 若 a 与 b 夹角 θ∈? ,π?时,则只要 a· b<0 且 b≠λa(λ<0). ?2 ? 当 a· b<0 有-2+2y<0 解得 y<1.② 由①②得 y<-1 或-1<y<1 【答案】 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; → 与CD → 的夹角的余弦值. (2)在(1)的条件下,求向量BC 【解】 → =AB → +BC → =(10,k+1),由题意知 A,C,D 三点共线, (1)因为AC (-∞,-1)∪(-1,1) → ∥CD → ,所以 10×1-2(k+1)=0,即 k=4. 所以AC → → BC·CD → → → (2) 因为 CD = (2 , 1) ,设向量 BC 与 CD 的夹角为 θ ,则 cos θ = = → ||CD →| |BC 12 3 10 = 10 . 4 2× 5 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|= |a+b|= k2+(k+2)2, 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°= 1 即-2= (ka-b)· (a+b) |ka-b||a+b| -2 , , 2· k2+(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 B.?-3,-9? ? ? 7? ? 7 D.?-9,-3? ? ? 设 c=(x,y), ?7 7? A.?9,3? ? ? ?7 7? C.?3,9? ? ? 【解析】 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

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