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WWW.WWW.LHC123.COM:2018-2019学年高中数学苏教版必修四 课下能力提升:(二十二) 向量的应用-含答案

数学 课下能力提升(二十二) 向量的应用 一、填空题 1.已知△ABC 中, AB =a, AC =b,若 a·b<0,则△ABC 的形状为________. 2.过点 A(2,3)且垂直于向量 a=(2,1)的直线方程为____________________. 3.作用于原点的两个力 F1=(1,1),F2(2,3),为使它们平衡,需要加力 F3=________. 4.当两人提起重量为|G|的旅行包时,两人用力都为|F|,夹角为 θ ,若|F|=|G|,则 θ 的 值为________. 5.已知直线 ax+by+c=0 与圆 x +y =1 相交于 A、B 两点,且|AB|= 3,则 OA · OB = 2 2 ??? ? ??? ? ??? ? ??? ? __________. 二、解答题 6.如图,在平行四边形 ABCD 中,已知 AD=1,AB=2,对角线 BD=2,求对角线 AC 的长. 7.在直角三角形 ABC 中,AB=4,AC=3,∠A=90°,CD 是直角三角形 ABC 的角平分线, 求 CD 的长. 数学 8.某人骑车以每小时 a 公里的速度向东行驶,感到风从正北方向吹来,而当速度为 2a 公里 /小时时,感到风从东北方向吹来,试求实际风速和方向. 答 案 数学 1.解析:由 a·b<0? ∠A>90°,故为钝角三角形. 答案:钝角三角形 2.解析:设 P(x,y)是所求直线上的任意一点(A 点除外), 则 AP ⊥a,∴ AP ·a=0.又∵ AP =(x-2,y-3). ∴2(x-2)+(y-3)=0,即 2x+y-7=0. 又∵点 A(2,3)在直线 2x+y-7=0 上, ∴所求直线方程为 2x+y-7=0. 答案:2x+y-7=0 ? ?1+2+x=0, 3. 解析: 要使它们平衡, 则合力大小为 0, F1+F2+F3=0, 设 F3=(x, y), 则? ?1+3+y=0, ? ??? ? ??? ? ??? ? 解得? ?x=-3, ? ? ?y=-4, 故 F3=(-3,-4). 答案:(-3,-4) 4.解析:作 OA =F1, OB =F2, OC =-G,则 OC = OA + OB , 当|F1|=|F2|=|G|时,△OAC 为正三角形, ∴∠AOC=60°,从而∠AOB=120°. 答案:120° 5.解析:如图,取 D 为 AB 的中点, ∵OA=1,AB= 3,∴∠AOD= 2π ∴∠AOB= . 3 ∴ OA · OB =1×1×cos 1 答案:- 2 6.解:设 AD =a, AB =b,则 AC =a+b, BD =a-b, 由已知|a|=1,|b|=2,|a-b|=2. 则(a-b) =|a-b| =4,即 a -2a·b+b =4, 1 则 1-2a·b+4=4,所以 a·b= . 2 1 2 2 2 2 所以|a+b| =(a+b) =a +2a·b+b =1+2× +4=6, 2 即|a+b|= 6. 故| AC |= 6,即对角线 AC 的长为 6. 2 2 2 2 ??? ? ??? ? ??? ? ??? ? ??? ? ??? ? π . 3 ??? ? ??? ? 2π 1 =- . 3 2 ??? ? ??? ? ??? ? ??? ? ??? ? 数学 7.解:以 A 为原点,AB 所在的直线为 x 轴,AC 所在的直线为 y 轴,建立平面直角坐标系, 则 A(0,0), B(4,0), C(0,3), 则 CA =(0, -3), -3), | CA |=3, | CB |= 4 +?-3? CB =(4, 2 ??? ? ??? ? ??? ? ??? ? 2 =5. 设 D(x,0),则 CD =(x,-3), 又∵CD 是直角三角形 ABC 的角平分线, ??? ? ??? ? ??? ? ??? ? ? CA CB ? ? + ??? ? ?=λ ∴ CD =λ ? ??? ?| CA | | CB |? ??0,-3?+?4,-3?? ? ? 3 5 ? ? 3 ? ?x=2, 解得? 15 ?λ = 8 , ? 4λ =x, ? ? 5 4λ 8λ ? ? =? ,- ?,∴? 5 ? ? 5 8λ ?- 5 =-3, ? ??? ? 3 ? CD =? ?2,-3?, ? ? 故 CD = ??? ? ?3?2+?-3?2=3 5,∴CD 的长为3 5. ?2? 2 2 ? ? 8.解:设 a 表示此人以每小时 a 公里的速度向东行驶的向量,无风时此人感到风速为-a. 设实际风速为 v, 那么此人感到的风速为 v-a, 如图所示, 设 OA ??? ? =-a, ??? ? OB =-2a, ??? ? ??? ? ??? ? ??? ? ∵ PO + OA = PA ,∴ PA =v-a, 这就是感到由正北方向吹来的风速. ∵ PO + OB = PB , ∴ PB =v-2a, 于是当此人的速度是原来的 2 倍时所感受到由东北方 向吹来的风速就是 PB . 由题意:∠PBO=45°,PA⊥BO,BA=AO, 从而△POB 为等腰直角三角形,∴PO=PB= 2a,即|v|= 2a, ∴实际为风速是 2a 的西北风. ??? ? ??? ? ??? ? ??? ? ??? ?

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