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question_answer1) Let \[\vec{a}=\hat{i}-\hat{j},\,\,\vec{b}=\hat{i}+\hat{j}+\hat{k}\] and \[\vec{c}\] be a vector such that \[\vec{a}\times \vec{c}+\vec{b}=\vec{0}\] and \[\vec{a}\,\,.\,\,\vec{c}=4,\] then \[|\vec{c}{{|}^{2}}\] is equal to:
question_answer2) A particle acted on by constant forces \[\vec{f}=4\hat{i}+3\hat{j}-3\hat{k}\] and \[\vec{g}=3\hat{i}+\hat{j}-\hat{k}\] experiences a displacement from the point \[\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\] to the point \[\vec{b}=5\hat{i}+4\hat{j}+\hat{k}.\] The total work done by the forces is
question_answer3) Let \[\vec{a}=\hat{i}+\hat{j}+\sqrt{2}\,\hat{k},\] \[\vec{b}={{b}_{1}}\,\hat{i}+{{b}_{2}}\hat{j}+\sqrt{2}\hat{k}\] and \[\vec{c}=5\hat{i}+\hat{j}+\sqrt{2}\hat{k}\] be three vectors such that the projection vector of \[\vec{b}\] on \[\vec{a}\] is \[\vec{a}\]. If \[\vec{a}+\vec{b}\] is perpendicular to \[\vec{c},\] then \[\left| {\vec{b}} \right|\] is equal to
question_answer4) Let \[\vec{u}=\hat{i}+\hat{j},\,\,\,\vec{v}=\hat{i}-\hat{j}\] and \[\vec{w}=\hat{i}+2\hat{j}+3\hat{k}.\] If \[\hat{n}\] is a unit vector such that \[\vec{u}.\hat{n}=0\] and \[\vec{v}.\hat{n}=0,\] then \[\left| \vec{w}.\hat{n} \right|\] is equal to
question_answer5) Let \[\overset{\to }{\mathop{\alpha }}\,=(\lambda -2)\,\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,\] and \[\overset{\to }{\mathop{\beta }}\,=(4\lambda -2)\,\overset{\to }{\mathop{a}}\,+3\overset{\to }{\mathop{b}}\,\]be two given vectors where vectors \[\overset{\to }{\mathop{a}}\,\] and \[\overset{\to }{\mathop{b}}\,\] are non-collinear. The value of \[\left| \lambda \right|\] for which vectors \[\overset{\to }{\mathop{\alpha }}\,\] and \[\overset{\to }{\mathop{\beta }}\,\] are collinear, is
question_answer6) If \[\vec{x}\] and \[\vec{y}\] are two unit vectors and \[\pi \] is the angle between them, then \[\frac{1}{2}|\vec{x}-\vec{y}|\] is equal to
question_answer7) Let \[\sqrt{3}\hat{i}+\hat{j},\,\,\hat{i}+\sqrt{3}\hat{j}\] and \[\beta \hat{i}+(1-\beta )\hat{j}\] respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is \[\frac{3}{\sqrt{2}},\] then the sum of all possible values of \[\beta \] is
question_answer8) The vectors \[\overrightarrow{AB}=3\hat{i}+5\hat{j}+4\hat{k}\] and \[\overrightarrow{AC}=5\hat{i}-5\hat{j}+2\hat{k}\] are the sides of a triangle ABC. The length of the median through A is
question_answer9) Let \[\vec{a},\,\vec{b}\] and \[\vec{c}\] be three unit vectors, out of which vectors \[\vec{b}\] and \[\vec{c}\] are non-parallel. If \[\alpha \] and \[\beta \] are the angles which vector \[\vec{a}\] makes with vectors \[\vec{b}\] and \[\vec{c}\] respectively and then \[|\alpha -\beta |\] is equal to
question_answer10) If \[\vec{a},\vec{b},\vec{c}\] are vectors such that \[[\vec{a}\,\vec{b}\,\vec{c}]=4\] then \[[\vec{a}\times \vec{b}\,\,\,\,\vec{b}\times \vec{c}\,\,\,\,\vec{c}\times \vec{a}]=\]
question_answer11) Let \[\alpha \in R\] and the three vectors \[\overset{\to }{\mathop{a}}\,=\alpha \hat{i}+\hat{j}+3\hat{k},\] \[\overset{\to }{\mathop{b}}\,=2\hat{i}+\hat{j}-\alpha \hat{k}\] and \[\overset{\to }{\mathop{c}}\,=\alpha \hat{i}-2\hat{j}+3\hat{k}.\] Then, the number of elements in the set S=( \[\alpha :\overset{\to }{\mathop{a}}\,,\,\,\vec{b}\] and \[\overset{\to }{\mathop{c}}\,\] are coplanar) is
question_answer12) If and vectors \[(1,\,\,a,\,\,{{a}^{2}}),\] \[(1,b,{{b}^{2}})\] and \[(1,c,{{c}^{2}})\] are non-coplanar, then the value of \[abc+1\]is the product \[abc\] equals
question_answer13) If \[|\vec{a}|\,=5,\,\,|\vec{b}|\,=4,\,\,|\vec{c}|\,=3\] and \[\vec{a}+\,\vec{b}+\,\vec{c}=0\] then the value of \[|\vec{a}\cdot \vec{b}\,+\vec{b}\cdot \vec{c}+\,\vec{c}\cdot \vec{a}|,\] is
question_answer14) If \[\vec{a},\vec{b},\vec{c}\] are vectors such that \[\vec{a}+\vec{b}+\vec{c}=0\] and \[|\vec{a}|=7,\] \[|\vec{b}|=5,\] \[|\vec{c}|\,=3\] then angle between vector \[\vec{b}\] and \[\vec{c}\] is
question_answer15) Let \[\bar{u},\,\bar{v},\bar{w}\] be such that \[|\bar{u}|=1,\] \[|\bar{v}|=2,\] \[|\bar{w}|=3.\]. If the projection \[\bar{v}\] along \[\bar{u}\] is equal to that of \[\bar{w}\] along \[\bar{u}\] and \[\bar{v}\], \[\bar{w}\] are perpendicular to each other then \[|\bar{u}-\bar{v}+\bar{w}{{|}^{2}}\] equals
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