單元10 基礎題類題

  1. 下圖為一正立方體,若點\(A\)為原點,點\(B\)之坐標為為\((\,1\;,\;0\;,\;0\,)\),點\(D\)之坐標為\((\,0\;,\;1\;,\;0\,)\),點\(E\)之坐標為\((\,0\;,\;0\;,\;1\,)\),則以任兩頂點為起點與終點所構成之向量中,與向量\((\, - 1\;,\;0\;,\;1\,)\)相等的有__________個。
    10 01
  1. 若四邊形\(ABCD\)為一梯形,其中點\(A\)之坐標為\((\,1\;,\;2\,)\),點\(B\)之坐標為\((\, - 2\;,\;0\,)\),點\(C\)之坐標為\((\,1\;,\;k\,)\),點\(D\)之坐標為\((\,2\;,\;5\,)\),則\(k\)之值為__________。
  1. 已知空間上相異四點\(A\)、\(B\)、\(C\)、\(D\),且\(O\)為原點,若\(\overset{\rightharpoonup} {AC} = (\,9\;,\;2\;,\;2\,)\), \(\overset{\rightharpoonup} {BC}  = (\,3\;,\;4\;,\;5\,)\),\(\overset{\rightharpoonup} {OD}  = t\overset{\rightharpoonup} {AB} \),其中\(t\)為實數,則:
    (1)_ \(\overset{\rightharpoonup} {AB} \)=__________。
    (2)_ 若\(|\overset{\rightharpoonup} {OD} | = 14\),則\(t = \)__________。
    (3)_ 點\(D\)之坐標為__________。
  1.  下圖中每一個小三角形皆為正三角形,設\(\overset{\rightharpoonup} {AB}  = \overset{\rightharpoonup} a \) ,\(\overset{\rightharpoonup} {AC}  = \overset{\rightharpoonup} b \)。請回答下列問題:
    10 04
    (1) 若\(\overset{\rightharpoonup} {BC}  = x\overset{\rightharpoonup} a  + y\overset{\rightharpoonup} b \) ,則數對
       \((\,x\;,\;y\,) = \)__________。
    (2) 若\(\overset{\rightharpoonup} {DE}  = r\overset{\rightharpoonup} a  + s\overset{\rightharpoonup} b \) ,則數對
       \((\,r\;,\;s\,) = \)__________。
  2. 設空間中點\(P\)在直線\(AB\)上,且\(3\overline {AP}  = 4\overline {PB} \),\(O\)為空間中一點,若
    \(\overset{\rightharpoonup} {OP}  = x\overset{\rightharpoonup} {OA}  + y\overset{\rightharpoonup} {OB} \) ,則數對\((\,x\;,\;y\,) = \)__________。
  3. 坐標空間中\(A(\,2\;,\;0\;,\;a\,)\),\(B(\, - 6\;,\;b\;,\;3\,)\),\(C(\,c\;,\;3\;,\; - 5\,)\)三點共線。已知\(C\)不在\(A\),\(B\)之間,且\(\overline {AC} :\overline {BC}  = 3:4\),則\(a + b + c = \)__________。
  4. 若\(A(\,1\;,\;0\;,\;5\,)\),\(B(\,3\;,\;a\;,\;2\,)\),\(C(\,5\;,\;6\;,\;b\,)\)三點共線,則數對\((\,a\;,\;b\,) = \)__________。
  5. __________\(O\)、\(A\)、\(B\)、\(P\)為平面上相異四點,下列哪些條件使得點\(P\)必在直線\(AB\)上?(多選)  (1) \(\overset{\rightharpoonup} {AB}  = \overset{\rightharpoonup} {PB}  - \overset{\rightharpoonup} {PA} \)  (2) \(\overset{\rightharpoonup} {AB}  = 3\overset{\rightharpoonup} {PB} \)
    (3) \(\overset{\rightharpoonup} {OP}  = \frac{1}{3}\overset{\rightharpoonup} {OA}  + \frac{2}{3}\overset{\rightharpoonup} {OB} \)  (4) \(2\overset{\rightharpoonup} {OP}  = 2\overset{\rightharpoonup} {OA}  - \overset{\rightharpoonup} {OB} \)  (5) \(7\overset{\rightharpoonup} {OP}  = 9\overset{\rightharpoonup} {OA}  - 2\overset{\rightharpoonup} {OB} \)
  6. 在坐標平面上,\(\Delta ABC\)內有一點\(P\)滿足\(\overset{\rightharpoonup} {AP}  = (\,\frac{3}{4}\;,\;\frac{5}{6}\,)\)及
    \(\overset{\rightharpoonup} {AP}  = \frac{1}{3}\overset{\rightharpoonup} {AB}  + \frac{1}{5}\overset{\rightharpoonup} {AC} \) 。若\(A\),\(P\)連線交\(\overline {BC} \)於\(M\),則\(\overset{\rightharpoonup} {AM} \)=__________。
  7. \(\Delta ABC\)中,\(\overline {AB}  = 6\),\(\overline {BC}  = 8\),\(\overline {CA}  = 10\),則:
    (1) \(\overset{\rightharpoonup} {AB}  \cdot \overset{\rightharpoonup} {AC} \)=__________。
    (2) \(\overset{\rightharpoonup} {AB}  \cdot \overset{\rightharpoonup} {BC} \)=__________。
    (3) \(\overset{\rightharpoonup} {BC}  \cdot \overset{\rightharpoonup} {CA} \)=__________。
  8. \(\Delta ABC\)中,\(\overline {AB}  = 7\),\(\overline {BC}  = 8\),\(\overline {CA}  = 9\),則\(\overset{\rightharpoonup} {AB}  \cdot \overset{\rightharpoonup} {BC} \)=__________。
  9. \(\Delta ABC\)中,\(A(\,1\;,\;2\,)\),\(B(\, - 2\;,\;6\,)\),\(C(\,13\;,\;7\,)\),則:
    (1) \(\overset{\rightharpoonup} {AB}  \cdot \overset{\rightharpoonup} {AC} \)=__________。
    (2) \(\cos A = \)__________。
  10. 設\(\overset{\rightharpoonup} a \)與\(\overset{\rightharpoonup} b \)兩向量之長各為\(\frac{3}{2}\)與5,若\(\overset{\rightharpoonup} a \)與\(\overset{\rightharpoonup} b \)之夾角為\({120^ \circ }\),則|\(2\overset{\rightharpoonup} a  - \overset{\rightharpoonup} b \)|=__________。
  11. 設\(\overset{\rightharpoonup} u  = (3,2)\),\(\overset{\rightharpoonup} v  = (2, - 1)\), \(\overset{\rightharpoonup} w  = \overset{\rightharpoonup} u  + t\overset{\rightharpoonup} v \) ,其中\(t\)為實數,則當\(t = \)__________時,\(\overset{\rightharpoonup} w \)的長度有最小值為__________。
  12. 設\(A(\, - 2\;,\; - 2\;,\;1\,)\),\(B(\,0\;,\; - 1\;,\; - 3\,)\),\(C(\, - 1\;,\; - 2\;,\; - 1\,)\),則:
    (1) \(\overset{\rightharpoonup} {AB} \)在\(\overset{\rightharpoonup} {AC} \)上的正射影為__________。
    (2) 點\(B\)在直線\(AC\)上的投影點坐標為__________。
  13. 已知\(x\),\(y\),\(z\)為實數,若\({(x - 1)^2} + {(y - 1)^2} + {(z - 1)^2} = 4\),則\(2x + y - 2z\)之最大值為__________,此時序組\((\,x\;,\;y\;,\;z\,) = \)__________;最小值為__________,此時序組\((\,x\;,\;y\;,\;z\,) = \)__________。
  14. 已知\(x\),\(y\),\(z\)為實數,若\(2x + y - 2z = 14\),則當序組\((\,x\;,\;y\;,\;z\,) = \)__________時,\({x^2} + {y^2} + {z^2} - 2x + 6y\)有最小值__________。

 

Ans:

  1. 2
  2. 9或\(\frac{{13}}{3}\)
  3. (1) \((\,6\;,\; - 2\;,\; - 3\,)\) (2) \( \pm 2\) (3) \((\,12\;,\; - 4\;,\; - 6\,)\)或\((\, - 12\;,\;4\;,\;6\,)\)
  4. (1) \((\, - 1\;,\;1\,)\) (2) \((\, - 3\;,\;2\,)\)
  5. \((\,\frac{3}{7}\;,\;\frac{4}{7}\,)\)或\((\, - 3\;,\;4\,)\)
  6. 26
  7. \((\,3\;,\; - 1\,)\)
  8. (2)(3)(5)
  9. \((\,\frac{{45}}{{32}}\;,\;\frac{{25}}{{16}}\,)\)
  10. (1) \(30\sqrt 3 \) (2) 0 (3) 40
  11. \( - 16\)
  12. (1) \( - 16\) (2) \( - \frac{{16}}{{65}}\)
  13. 7
  14. \( - \frac{4}{5}\),\(\frac{{7\sqrt 5 }}{5}\)
  15. (1) \((\,2\;,\;0\;,\; - 4\,)\) (2) \((\,0\;,\; - 2\;,\; - 3\,)\)
  16. 7,\((\,\frac{7}{3}\;,\;\frac{5}{3}\;,\; - \frac{1}{3}\,)\);\( - 5\),\((\, - \frac{1}{3}\;,\;\frac{1}{3}\;,\;\frac{7}{3}\,)\)
  17. \((\,\frac{{13}}{3}\;,\; - \frac{4}{3}\;,\; - \frac{{10}}{3}\,)\),\( - 1\)

pdf檔請至資源分享中下載,謝謝

Log in to comment

Share this post

Submit to FacebookSubmit to Google PlusSubmit to Twitter