优优班--学霸训练营 > 知识点挑题
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            • 1.

              已知平面上直线\(l\)的方向向量\(e=(- \dfrac{4}{5}, \dfrac{3}{5}) \),点\(O(0,0)\)和\(A(1,-2)\)在\(l\)上的射影分别是\(O′\)和\(A′\),则\(\overrightarrow{{O}{{{"}}}{A}{{{"}}}}=\lambda e\),其中\(λ\)等于  \((\)    \()\)

              A.\(\dfrac{{11}}{{5}}\)
              B.\(-\dfrac{11}{{5}}\)
              C.\(2\)
              D.\(-2\)
            • 2. 已知向量 \(a\)\(b\)满足\(|\) \(b\)\(|=4\), \(a\)\(b\)方向上的投影是\( \dfrac{1}{2}\),则 \(a\)\(·\) \(b\)\(=\)________.
            • 3.

              若非零向量\(\overrightarrow{m}\),\(\overrightarrow{n}\)的夹角为锐角\(θ\),且\(\dfrac{|\overrightarrow{m}|}{|\overrightarrow{n}|}=\cos \beta \),则称\(\overrightarrow{m}\)被\(\overrightarrow{n}\)“同余”\(.\)已知\(\overrightarrow{b}\)被\(\overrightarrow{a}\)“同余”,则\(\overrightarrow{a}-\overrightarrow{b}\)在\(\overrightarrow{a}\)上的投影是

              A.\(\dfrac{{{\overrightarrow{a}}^{2}}-{{\overrightarrow{b}}^{2}}}{|\overrightarrow{b}|}\)
              B.\(\dfrac{{{\overrightarrow{a}}^{2}}-{{\overrightarrow{b}}^{2}}}{{{\overrightarrow{a}}^{2}}}\)
              C.\(\dfrac{{{\overrightarrow{b}}^{2}}-{{\overrightarrow{a}}^{2}}}{|\overrightarrow{a}|}\)
              D.\(\dfrac{{{\overrightarrow{a}}^{2}}-{{\overrightarrow{b}}^{2}}}{|\overrightarrow{a}|}\)
            • 4.

              若非零向量\(a\),\(b\)的夹角为锐角\(θ\),且\( \dfrac{|a|}{|b|}=\cos θ\),则称\(a\)被\(b\)“同余”\(.\)已知\(b\)被\(a\)“同余”,则\(a-b\)在\(a\)上的投影是\((\)  \()\)

              A.\( \dfrac{a^{2}-b^{2}}{|a|}\)
              B.\( \dfrac{a^{2}-b^{2}}{a^{2}}\)
              C.\( \dfrac{b^{2}-a^{2}}{|a|}\)
              D.\( \dfrac{a^{2}-b^{2}}{|b|}\)
            • 5.

              已知向量\(a=(2,m)\),\(b=(-1,2)\),若\(a⊥b\),\(c=a+b\),则\(a\)在向量\(c\)方向上的投影为              \(.\) 

            • 6.

              已知的外接圆圆心为\(O\),半径为\(R\),三个内角\(A\),\(B\),\(C\)成等差数列,向量\( \overrightarrow{BA} \)在向量\( \overrightarrow{BC} \)的方向上的投影为\(1\),则\( \overrightarrow{AO}· \overrightarrow{AB} =\)(    )

              A.\(2R\)
              B.\(R\)
              C.\(2\)
              D.\(-2\)
            • 7. 已知平面上直线 \(l\)\(e\)所在直线平行且 \(e\)\(=\left(\begin{matrix} \begin{matrix}- \dfrac{4}{5}, \dfrac{3}{5} \end{matrix}\end{matrix}\right)\),点 \(O\)\((0,0)\)和 \(A\)\((1,-2)\)在 \(l\)上的射影分别是 \(O\)\(′\)和 \(A\)\(′\),则\(\overrightarrow{O\prime A\prime }=\) \(λe\),其中 \(λ\)等于(    )
              A.\( \dfrac{11}{5}\)                                   
              B.\(- \dfrac{11}{5}\)
              C.\(2\)                                                
              D.\(-2\)
            • 8.

              已知两个不共线的单位向量\(e_{1}\),\(e_{2}\)的夹角为 \(θ\),则下列结论不一定正确的是\(( \) \()\)

              A.\(e_{1}\)在\(e_{2}\)方向上的投影为\(\cos \) \(θ\)
              B.\(e_{1}·e_{2}\) \(=\)\(1\)
              C. \(e_{1}^{2}=e_{2}^{2} \)
              D.\((e_{1} \)\(+\)\(e_{2})⊥(e_{1}\) \(-\)\(e_{2})\)
            • 9.

              \((1) \overset{⇀}{a} =(\)\(x\)\(-1\),\(y\)\()\),\( \overset{⇀}{b} =(1,2)\),且\(\overrightarrow{a}\bot \overrightarrow{b}\),则当\(x > 0,y > 0\)时,\(\dfrac{1}{x}+\dfrac{1}{y}\)的最小值为 ______.

              \((2)\)向量\(\overrightarrow{a}=\left( 2,3 \right)\)在\(\overrightarrow{b}=\left( -4,7 \right)\)上的投影是 ______.

              \((3)\)若等比数列\(\left\{ {{a}_{n}} \right\}\)满足\({{a}_{1}}+{{a}_{3}}=20\),\({{a}_{2}}+{{a}_{4}}=40\),则公比\(q=\) ______\(..\)

              \((4)\)已知\(m,n,p\)表示不重合的三条直线,表示不重合的三个平面\(.\)给出以下四个命题

              \(①\)若\(m⊥p\),\(m/\!/n\)则\(n⊥p\)
              \(②\)若\(m/\!/β\),\(n/\!/β\),\(m⊂ α\),\(n⊂ α\),则\(α/\!/β③\)若\(α⊥γ\),\(β⊥γ\),\(α∩β=m\),则\(m⊥γ\)
              \(④\)若\(α/\!/β\),\(m⊂ α\),\(n⊂ β\),则\(m/\!/n\)

              其中正确命题的序号是          \((\)把所有正确命题的序号都写上\()\) 

            • 10.

              在\(\Delta ABC\)中,\(| \overset{→}{AB}+ \overset{→}{AC}|=| \overset{→}{AB}− \overset{→}{AC}| \),\(AB=3\),\(AC=4\),则\(\overrightarrow{BC}\)在\(\overrightarrow{CA}\)方向上的投影是

              A.\(4\)             
              B.\(3\)                 
              C.\(-4\)
              D.\(-3\)
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