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67847.com:18学年高中数学第25课时平面向量的数量积的坐标表示、模、夹角练习新人教A版必修4

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第 25 课时

平面向量的数量积的坐标表示、模、夹角

课时目标 1.掌握向量数量积的坐标表示,会进行向量数量积的坐标运算. 2.会用坐标运算求向量的模,并会用坐标运算判断两个向量是否垂直. 3.能运用数量积的坐标求出两个向量夹角的余弦值. 识记强化 1.若 a=(x1,y1),b=(x2,y2),则 a·b=x1x2+y1y2. → → 2.若有向线段AB,A(x1,y1),B(x2,y2),则|AB|= x2-x1 2+ y2-y1 → 2 2 (x,y),则|AB|= x +y . 3.若 a=(x1,y1),b=(x2,y2),则 a⊥b?x1x2+y1y2=0. 4.两向量 a=(x1,y1),b=(x2,y2),则求两向量的夹角 θ 的公式为 x1x2+y1y2 cosθ = 2 . 2 2 2 x1+y1 · x2+y2

2

→ ;若AB=

课时作业

一、选择题

1.设向量 a=(x,1),b=(4,x),且 a⊥b,则 x 的值是( ) A.±2 B.0 C.-2 D.2 答案:B 解析:由 a⊥b,得 a·b=0,即 4x+x=0,解得 x=0,故选 B. 2.已知向量 a=(0,-2 3),b=(1, 3),则向量 a 在 b 方向上的投影为( ) A. 3 B.3 C.- 3 D.-3 答案:D a·b -6 解析:向量 a 在 b 方向上的投影为 = =-3.选 D. |b| 2 3.已知向量 a=(k,3),b=(1,4),c=(2,1),且(2a-3b)⊥c,则实数 k 的值为( ) 9 A.- B.0 2 15 C.3 D. 2 答案:C 解析:∵2a-3b=(2k-3,-6).又(2a-3b)⊥c,∴(2a-3b)·c=0,即(2k-3)×2 +(-6)=0,解得 k=3. 4.若 A(1,2),B(2,3),C(-3,5),则△ABC 为( )

1

A.直角三角形 B.锐角三角形 C.钝角三角形 D.不等边三角形 答案:C 解析:∵A(1,2),B(2,3),C(-3,5), → → ∴AB=(1,1),AC=(-4,3), → → AB·AC - +1×3 1 cosA= = =- <0,∴∠A 为钝角,△ABC 为钝角三角 → → 2× 25 5 2 |AB||AC| 形. 5.若向量 a=(x+1,2) 和向量 b=(1,-1)平行,则|a+b|=( ) 10 A. 10 B. 2 C. 2 D. 2 2

答案:C 解析:由题意得,-(x+1)-2×1=0 得 x=-3.故 a+b=(-1,1). 2 2 ∴|a+b|= - + - = 2 → → 6.如图,在等腰直角三角形 AOB 中,设OA=a,OB=b,OA=OB=1,C 为 AB 上靠近点 A → 的四等分点,过 C 作 AB 的垂线 l,设 P 为垂线上任意一点,OP=p,则 p·(b-a)=( )

1 1 A.- B. 2 2 3 3 C.- D. 2 2 答案:A → → 解析:因为在等腰直角三角形 AOB 中,OA=a,OB=b,OA=OB=1,所以|a|=|b|=1, a·b=0. 1 1 → 由题意,可设OP=- (b-a)+λ · (b+a),λ ∈R, 4 2 所以 p·(b-a) 1 λ =- (b-a)·(b-a)+ (b+a)·(b-a) 4 2 1 λ 2 2 2 =- (b-a) + (|b| -|a| ) 4 2 1 2 2 =- (|a| +|b| -2a·b) 4 1 =- (1+1-0) 4 1 =- . 2 二、填空题 7.已知 a=(1,2),b=(x,4),且 a·b=10,则|a-b|=________. 答案: 5
2

解析:由题意,得 a·b=x+8=10,∴x=2,∴a-b=(-1,-2),∴|a-b|= 5. → → 8.已知点 A(4,0),B(0,3),OC⊥AB 于点 C,O 为坐标原点,则OA·OC=________. 144 答案: 25 解析:设点 C 的坐标为(x,y),因为 OC⊥AB 于点 C, → → ? ?OC·AB=0 ∴? → → ? ?AC∥AB 即?
? ?

, -4, =-4x+3y=0

x,y

?3x+4y-12=0 ?



36 ? ?x=25 解得? 48 ? ?y=25

144 → → ,∴OA·OC=4x= . 25

9.若平面向量 a=(log2x,-1),b=(log2x,2+log2x),则满足 a·b<0 的实数 x 的取 值集合为________. ? 1 ? ? ? ? 答案:?x? <x<4 ? ? ? ? ?2 ? 解析:由题意可得(log2x) -log2x-2<0? (log2x+1)(log2x-2)<0,所以-1<log2x<2, 1 所以 <x<4. 2 三、解答题 → → → 10.已知 O 为坐标原点,OA=(2,5),OB=(3,1),OC=(6,3),则在线段 OC 上是否存在 → → 点 M,使得MA⊥MB?若存在,求出点 M 的坐标;若不存在,请说明理由. → → 解:假设存在点 M,且OM=λ OC=(6λ ,3λ )(0≤λ ≤1), → → ∴MA=(2-6λ ,5-3λ ),MB=(3-6λ ,1-3λ ). → → ∵MA⊥MB, ∴(2-6λ )(3-6λ )+(5-3λ )(1-3λ )=0, 1 11 2 即 45λ -48λ +11=0,解得 λ = 或 λ = . 3 15 → → ?22 11? ∴OM=(2,1)或OM=? , ?. ?5 5? ?22 11? ∴存在 M(2,1)或 M? , ?满足题意. ?5 5? π π 11.已知平面向量 a=(sinα ,1),b=(1,cosα ),- <α < . 2 2 (1)若 a⊥b,求 α ; (2)求|a+b|的最大值. 解:(1)由已知,得 a·b=0, 即 sinα +cosα =0,∴tanα =-1. π π π ∵- <α < ,∴α =- . 2 2 4 2 2 2 2 2 (2)由已知得|a+b| =a +b +2a·b=sin α +1+cos α +1+2(sinα +cosα )=3+ π? ? 2 2sin?α + ?. 4? ?
3
2

π π ∵- <α < , 2 2 π? π π 3π 2 ? 2 ∴- <α + < , ∴- <sin?α + ?≤1, 即 1<|a+b| ≤3+2 2, ∴1<|a+b|≤1 4? 4 4 4 2 ? + 2, 即|a+b|的最大值为 1+ 2.

能力提升

? π π ? 则|a+b|的取值范围是( 12. 若 a=(1,0), b=(cosθ , sinθ ), θ ∈?- , ?, ? 2 2?

)

A.[0, 2] B.[0, 2) C.[1,2] D.[ 2,2] 答案:D 2 2 2 2 解析:|a+b| =(a+b) =a +2a·b+b =2+2cosθ =2(1+cosθ ) ? π π? ∵θ ∈?- , ?,∴cosθ ∈[0,1]. ? 2 2? ∴2≤2(1+cosθ )≤4. ∴ 2≤|a+b|≤2. 1 3 2 13.已知 a=( 3,-1),b=( , ),且存在实数 k 和 t,使得 x=a+(t -3)b,y 2 2 k+t2 =-ka+tb,且 x⊥y,试求 的最小值.

t

解:由题知,|a|=2,|b|=1, 1 3 a·b= 3× -1× =0,∴a⊥b. 2 2 2 由 x⊥y 得,[a+(t -3)b]·(-ka+tb)=0, 2 3 2 2 即-ka +(t -3t)b +(t-t k+3k)a·b=0, 2 3 2 ∴-k|a| +(t -3t)b =0. t3-3t ∵|a|=2,|b|=1,∴k= . 4 k+t2 1 2 1 7 2 ∴ = (t +4t-3)= (t+2) - . t 4 4 4 k+t2 7 即当 t=-2 时, 有最小值- . t 4

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