單元10 基礎題類題
- 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。
- 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__________。
- 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__________。
- 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:
(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
\((\,r\;,\;s\,) = \)__________。 - 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\((\,x\;,\;y\,) = \)__________。 - 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\(a + b + c = \)__________。
- 若\(A(\,1\;,\;0\;,\;5\,)\),\(B(\,3\;,\;a\;,\;2\,)\),\(C(\,5\;,\;6\;,\;b\,)\)三點共線,則數對\((\,a\;,\;b\,) = \)__________。
- __________\(O\)、\(A\)、\(B\)、\(P\)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) \(7\overset{\rightharpoonup} {OP} = 9\overset{\rightharpoonup} {OA} - 2\overset{\rightharpoonup} {OB} \) - 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\(\overset{\rightharpoonup} {AM} \)=__________。 - \(\Delta ABC\)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) \(\overset{\rightharpoonup} {BC} \cdot \overset{\rightharpoonup} {CA} \)=__________。 - \(\Delta ABC\)中,\(\overline {AB} = 7\),\(\overline {BC} = 8\),\(\overline {CA} = 9\),則\(\overset{\rightharpoonup} {AB} \cdot \overset{\rightharpoonup} {BC} \)=__________。
- \(\Delta ABC\)中,\(A(\,1\;,\;2\,)\),\(B(\, - 2\;,\;6\,)\),\(C(\,13\;,\;7\,)\),則:
(1) \(\overset{\rightharpoonup} {AB} \cdot \overset{\rightharpoonup} {AC} \)=__________。
(2) \(\cos A = \)__________。 - 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|\(2\overset{\rightharpoonup} a - \overset{\rightharpoonup} b \)|=__________。
- 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__________。
- 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__________。 - 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\((\,x\;,\;y\;,\;z\,) = \)__________。
- 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:
- 2
- 9或\(\frac{{13}}{3}\)
- (1) \((\,6\;,\; - 2\;,\; - 3\,)\) (2) \( \pm 2\) (3) \((\,12\;,\; - 4\;,\; - 6\,)\)或\((\, - 12\;,\;4\;,\;6\,)\)
- (1) \((\, - 1\;,\;1\,)\) (2) \((\, - 3\;,\;2\,)\)
- \((\,\frac{3}{7}\;,\;\frac{4}{7}\,)\)或\((\, - 3\;,\;4\,)\)
- 26
- \((\,3\;,\; - 1\,)\)
- (2)(3)(5)
- \((\,\frac{{45}}{{32}}\;,\;\frac{{25}}{{16}}\,)\)
- (1) \(30\sqrt 3 \) (2) 0 (3) 40
- \( - 16\)
- (1) \( - 16\) (2) \( - \frac{{16}}{{65}}\)
- 7
- \( - \frac{4}{5}\),\(\frac{{7\sqrt 5 }}{5}\)
- (1) \((\,2\;,\;0\;,\; - 4\,)\) (2) \((\,0\;,\; - 2\;,\; - 3\,)\)
- 7,\((\,\frac{7}{3}\;,\;\frac{5}{3}\;,\; - \frac{1}{3}\,)\);\( - 5\),\((\, - \frac{1}{3}\;,\;\frac{1}{3}\;,\;\frac{7}{3}\,)\)
- \((\,\frac{{13}}{3}\;,\; - \frac{4}{3}\;,\; - \frac{{10}}{3}\,)\),\( - 1\)
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