Mathematics
Vector Algebra and 3D Geometry
Previous Years Questions

## MCQ (More than One Correct Answer)

Let $\hat{\imath}, \hat{\jmath}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let \begin{aligned} & \vec{a}=3 \hat{... LetP_{1}$$and$$P_{2}$$be two planes given by$$ \begin{aligned} &P_{1}: 10 x+15 y+12 z-60=0 \\ &P_{2}:-2 x+5 y+4 z-20=0 \end{aligned} $$Which... Let$$S$$be the reflection of a point$$Q$$with respect to the plane given by$$ \vec{r}=-(t+p) \hat{\imath}+t \hat{\jmath}+(1+p) \hat{k} $$where... Let O be the origin and$$\overrightarrow {OA} = 2\widehat i + 2\widehat j + \widehat k$$and$$\overrightarrow {OB} = \widehat i - 2\widehat j + 2\...
Let $$\alpha$$2 + $$\beta$$2 + $$\gamma$$2 $$\ne$$ 0 and $$\alpha$$ + $$\gamma$$ = 1. Suppose the point (3, 2, $$-$$1) is the mirror image of t...
Let a and b be positive real numbers. Suppose $$PQ = a\widehat i + b\widehat j$$ and $$PS = a\widehat i - b\widehat j$$ are adjacent sides of a parall...
Let L1 and L2 be the following straight lines.$${L_1}:{{x - 1} \over 1} = {y \over { - 1}} = {{z - 1} \over 3}$$ and $${L_2}:{{x - 1} \over { - 3}} = ... Three lines$${L_1}:r = \lambda \widehat i$$,$$\lambda  \in $$R,$${L_2}:r = \widehat k + \mu \widehat j$$,$$\mu  \in $$R and$${L_3}:r =...
Let L1 and L2 denote the lines$$r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k)$$, $$\lambda$$$$\in$$ R and $$r = \mu (2\wide... Let P1 : 2x + y$$-$$z = 3 and P2 : x + 2y + z = 2 be two planes. Then, which of the following statement(s) is(are) TRUE? Let$$\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k$$be a unit vector in$${{R^3}}$$and$$\widehat w = {1 \over {\sqrt 6 }}\left...
Consider a pyramid $$OPQRS$$ located in the first octant $$\left( {x \ge 0,y \ge 0,z \ge 0} \right)$$ with $$O$$ as origin, and $$OP$$ and $$OR$$ alon...
In $${R^3},$$ consider the planes $$\,{P_1}:y = 0$$ and $${P_2}:x + z = 1.$$ Let $${P_3}$$ be the plane, different from $${P_1}$$ and $${P_2}$$, which...
In $${R^3},$$ let $$L$$ be a straight lines passing through the origin. Suppose that all the points on $$L$$ are at a constant distance from the two ...
Let $$\Delta PQR$$ be a triangle. Let $$\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP}$$ and $$\overrightarrow c = \overrightarrow {PQ... Let$$\overrightarrow x ,\overrightarrow y $$and$$\overrightarrow z $$be three vectors each of magnitude$$\sqrt 2 $$and the angle between each pa... Two lines$${L_1}:x = 5,{y \over {3 - \alpha }} = {z \over { - 2}}$$and$${L_2}:x = \alpha ,{y \over { - 1}} = {z \over {2 - \alpha }}$$are coplana... A line$$l$$passing through the origin is perpendicular to the lines$$$\,{l_1}:\left( {3 + t} \right)\widehat i + \left( { - 1 + 2t} \right)\wideha... If the straight lines $$\,{{x - 1} \over 2} = {{y + 1} \over k} = {z \over 2}$$ and $${{x + 1} \over 5} = {{y + 1} \over 2} = {z \over k}$$ are coplan... The vector (s) which is/are coplanar with vectors $${\widehat i + \widehat j + 2\widehat k}$$ and $${\widehat i + 2\widehat j + \widehat k,}$$ and per... Let $${\overrightarrow A }$$ be vector parallel to line of intersection of planes $${P_1}$$ and $${P_2}.$$ Planes $${P_1}$$ is parallel to the vector... Let $$a$$ and $$b$$ two non-collinear unit vectors. If $$u = a - \left( {a\,.\,b} \right)\,b$$ and $$v = a \times b,$$ then $$\left| v \right|$$ is Which of the following expressions are meaningful? The vector $$\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)$$ is Let $$\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k$$ and $$\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\wi... ## Numerical Let$$\alpha$$,$$\beta$$and$$\gamma$$be real numbers such that the system of linear equationsx + 2y + 3z =$$\alpha$$4x + 5y + 6z =$$\beta$$7x + ... Let$$\alpha$$,$$\beta$$and$$\gamma$$be real numbers such that the system of linear equationsx + 2y + 3z =$$\alpha$$4x + 5y + 6z =$$\beta$$7x + ... Let$$\overrightarrow u $$,$$\overrightarrow v $$and$$\overrightarrow w $$be vectors in three-dimensional space, where$$\overrightarrow u $$and ... Let$$\overrightarrow a = 2\widehat i + \widehat j - \widehat k$$and$$\overrightarrow b = \widehat i + 2\widehat j + \widehat k$$be two vectors. Co... Three lines are given by$$r = \lambda \widehat i,\,\lambda \in R$$,$$r = \mu (\widehat i + \widehat j),\,\mu \in R$$and$$r = v(\widehat i + \wid... Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and t... Let a and b be two unit vectors such that a . b = 0. For some x, y$$\in$$R, let $$\overrightarrow c = x\overrightarrow a + y\overrightarrow b + \... Suppose that$$\overrightarrow p ,\overrightarrow q $$and$$\overrightarrow r $$are three non-coplanar vectors in$${R^3}$$. Let the components of a... Let$$\overrightarrow a \,\,,\,\,\overrightarrow b $$and$$\overrightarrow c $$be three non-coplanar unit vectors such that the angle between every ... Consider the set of eight vectors$$V = \left\{ {a\widehat i + b\widehat j + c\widehat k:a,b.c \in \left\{ { - 1,1} \right\}} \right\}.$$Three non-c... If$$\overrightarrow a ,\overrightarrow b $$and$$\overrightarrow c $$are unit vectors satisfying$${\left| {\overrightarrow a - \overrightarrow b... Let $$\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat i + 2\wide... If$$\overrightarrow a $$and$$\overrightarrow b $$are vectors in space given by$$\overrightarrow a = {{\widehat i - 2\widehat j} \over {\sqrt 5 }... If the distance between the plane $$Ax-2y+z=d$$ and the plane containing the lines $${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}$$ and$...

Let O be the origin and let PQR be an arbitrary triangle. The point S is such that$$\overrightarrow{OP}$$ . $$\overrightarrow{OQ}$$ + $$\overrightarro... The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x + y$$-$$2z = 5 and 3x$$-$$6y$$-$$2z = 7 is If the triangle PQR varies, then the minimum value of cos(P + Q) + cos(Q + R) + cos(R + P) is |$$\overrightarrow{OX} \times \overrightarrow{OY}$$| = ? Let$$P$$be the image of the point$$(3,1,7)$$with respect to the plane$$x-y+z=3.$$Then the equation of the plane passing through$$P$$and contai... Match the following :$$\,\,\,\,\,\,\,\,\,\,\,\,$$Column$$I$$(A)$$\,\,\,\,$$In$${R^2},$$If the magnitude of the projection vector of... Match the following :$$\,\,\,\,\,\,\,\,\,\,\,\,$$Column$$I$$(A)$$\,\,\,\,$$In a triangle$$\Delta XYZ,$$let$$a, b,$$and$$c$$be t... From a point$$P\left( {\lambda ,\lambda ,\lambda } \right),$$perpendicular$$PQ$$and$$PR$$are drawn respectively on the lines$$y=x, z=1$$and$$...
match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists: $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$... Consider the lines$${L_1}:{{x - 1} \over 2} = {y \over { - 1}} = {{z + 3} \over 1},{L_2} : {{x - 4} \over 1} = {{y + 3} \over 1} = {{z + 3} \over 2}$... If $$\overrightarrow a$$ and $$\overrightarrow b$$ are vectors such that $$\left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {29}$$... The equation of a plane passing through the line of intersection of the planes $$x+2y+3z=2$$ and $$x-y+z=3$$ and at a distance $${2 \over {\sqrt 3 }}... The point$$P$$is the intersection of the straight line joining the points$$Q(2, 3, 5)$$and$$R(1, -1, 4)$$with the plane$$5x-4y-z=1.$$If$$S$$... Let$$\overrightarrow a = \widehat i + \widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$and$$\overrightarrow c... Match the statements given in Column -$$I$$ with the values given in Column-$$II.$$ $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$ (A) $$\,\,\,\,... Let$$P,Q,R$$and$$S$$be the points on the plane with position vectors$${ - 2\widehat i - \widehat j,4\widehat i,3\widehat i + 3\widehat j}$$and ... Equation of the plane containing the straight line$${x \over 2} = {y \over 3} = {z \over 4}$$and perpendicular to the plane containing the straight ... Two adjacent sides of a parallelogram$$ABCD$$are given by$$\overrightarrow {AB} = 2\widehat i + 10\widehat j + 11\widehat k$$and$$\,\overrighta... If the distance of the point $$P(1, -2, 1)$$ from the plane $$x+2y-2z$$$$\, = \alpha ,$$ where $$\alpha > 0,$$ is $$5,$$ then the foot of the per... Match the statement in Column-$$I$$ with the values in Column-$$II$$ $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$ (A)$$\,\,\,\,$$ A line fro... If $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c$$ and $$\overrightarrow d$$ are unit vectors such that $$\left( {\overrightarrow a ... Let$$P(3,2,6)$$be a point in space and$$Q$$be a point on the line$$$\widehat r = \left( {\widehat i - \widehat j + 2\widehat k} \right) + \mu \l...
A line with positive direction cosines passes through the point $$P(2, -1, 2)$$ and makes equal angles with the coordinate axes. The line meets the pl...
Let two non-collinear unit vectors $$\widehat a$$ and $$\widehat b$$ form an acute angle. A point $$P$$ moves so that at any time $$t$$ the position v...
The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $$\overrightarrow a \,,\,\overrightarrow b ,\overrighta... Consider the lines$${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}\,\,\,\,{L_2}:{{x - 2} \over 1} = {{y + 2} \over 2} = {{z - 3} \o...
Consider the lines $${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}\,\,\,\,{L_2}:{{x - 2} \over 1} = {{y + 2} \over 2} = {{z - 3} \o... Consider the lines$${L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}\,\,\,\,{L_2}:{{x - 2} \over 1} = {{y + 2} \over 2} = {{z - 3} \...
Consider three planes $${P_1}:x - y + z = 1$$$$${P_2}:x + y - z = 1$$$ $${P_3}:x - 3y + 3z = 2$$$Let $${L_1},$$ $${L_2},$$ $${L_3}$$ be the line... The minimum of distinct real values of $$\lambda ,$$ for which the vectors $$- {\lambda ^2}\widehat i + \widehat j + \widehat k,$$ $$\widehat i - {\l... Let$$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$be unit vectors such that$${\overrightarrow a + \overrightarrow b + \overrigh... Consider the planes $$3x-6y-2z=15$$ and $$2x+y-2z=5.$$ STATEMENT-1: The parametric equations of the line of intersection of the given planes are $$x=... Let the vectors$$\overrightarrow {PQ} ,\,\,\overrightarrow {QR} ,\,\,\overrightarrow {RS} ,\,\,\overrightarrow {ST} ,\,\,\overrightarrow {TU} ,$$and... A plane which is perpendicular to two planes$$2x - 2y + z = 0$$and$$x - y + 2z = 4,$$passes through$$(1, -2, 1).$$The distance of the plane from... Let$$\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$and$$\overrightarrow ... A variable plane at a distance of the one unit from the origin cuts the coordinates axes at $$A,$$ $$B$$ and $$C.$$ If the centroid $$D$$ $$(x, y, z)... If$$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$are three non-zero, non-coplanar vectors and$$\overrightarrow {{b_1}} = \overr... If $$\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1$$ and $$\overrightarrow a \... If the lines$${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}$$and$$\,{{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1}$$intersect, t... The unit vector which is orthogonal to the vector$$3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k $$and is coplanar with the vectors... The value of$$k$$such that$${{x - 4} \over 1} = {{y - 2} \over 1} = {{z - k} \over 2}$$lies in the plane$$2x -4y +z = 7,$$is The value of$$'a'$$so that the volume of parallelopiped formed by$$\widehat i + a\widehat j + \widehat k,\widehat j + a\widehat k$$and$$a\widehat... If $${\overrightarrow a }$$ and $${\overrightarrow b }$$ are two unit vectors such that $${\overrightarrow a + 2\overrightarrow b }$$ and $${5\overri... Let$$\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k $$and$$\overrightarrow W = \overrightarrow i + 3\overright... Let $$\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overrightarrow j + \left( {1 - x} \rig... If$$\overrightarrow a \,,\,\overrightarrow b $$and$$\overrightarrow c $$are unit vectors, then$${\left| {\overrightarrow a - \overrightarrow b }... If the vectors $$\overrightarrow a ,\overrightarrow b$$ and $$\overrightarrow c$$ form the sides $$BC,$$ $$CA$$ and $$AB$$ respectively of a triangl... Let the vectors $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ and $$\overrightarrow d$$ be such that $$\left( {\overrightarrow a \ti... If$$\overrightarrow a \,,\,\overrightarrow b $$and$$\overrightarrow c $$are unit coplanar vectors, then the scalar triple product$$\left[ {2\over... Let $$a=2i+j-2k$$ and $$b=i+j.$$ If $$c$$ is a vector such that $$a.$$ $$c = \left| c \right|,\left| {c - a} \right| = 2\sqrt 2$$ and the angle betw... Let $$a=2i+j+k, b=i+2j-k$$ and a unit vector $$c$$ be coplanar. If $$c$$ is perpendicular to $$a,$$ then $$c =$$ If $$a = i + j + k,\overrightarrow b = 4i + 3j + 4k$$ and $$c = i + \alpha j + \beta k$$ are linearly dependent vectors and $$\left| c \right| = \sq... For three vectors$$u,v,w$$which of the following expression is not equal to any of the remaining three? Let$$\overrightarrow a = \widehat i - \widehat j,\overrightarrow b = \widehat j - \widehat k,\overrightarrow c = \widehat k - \widehat i.$$If$$... If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ are non coplanar unit vectors such that $$\overrightarrow a \times \left( {\overright... Let$$\overrightarrow u ,\overrightarrow v $$and$$\overrightarrow w $$be vectors such that$$\overrightarrow u + \overrightarrow v + \overrightar... If $$\overrightarrow a ,$$ $$\overrightarrow b$$ and $$\overrightarrow c$$ are three non coplanar vectors, then $$\left( {\overrightarrow a + \ov... Let$$\overrightarrow p $$and$$\overrightarrow q $$be the position vectors of$$P$$and$$Q$$respectively, with respect to$$O$$and$$\left| {\ov... Let $$\alpha ,\beta ,\gamma$$ be distinct real numbers. The points with position vectors $$\alpha \widehat i + \beta \widehat j + \gamma \widehat k,... Let$$a, b, c$$be distinct non-negative numbers. If the vectors$$a\widehat i + a\widehat j + c\widehat k,\widehat i + \widehat k$$and$$c\widehat i... Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$$ be three non-coplanar vectors and $$\overrightarrow p ,\overrightarrow q ,\overright... The number of vectors of unit length perpendicular to vectors$$\overrightarrow a = \left( {1,1,0} \right)$$and$$\overrightarrow b = \left( {0,1,1... Let $$\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k$$ and $$\overrightarrow c = {c_1}i + {c_2}j +... The points with position vectors$$60i+3j,40i-8j,ai-52j$$are collinear if The volume of the parallelopiped whose sides are given by$$\overrightarrow {OA} = 2i - 2j,\,\overrightarrow {OB} = i + j - k,\,\overrightarrow {OC... For non-zero vectors $${\overrightarrow a ,\,\overrightarrow b ,\overrightarrow c },$$ $$\left| {\left( {\overrightarrow a \times \overrightarrow b }... The scalar$$\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \ove... ## Subjective Match the statements / expressions given in Column-$$I$$ with the values given in Column-$$II.$$ $$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$ ... Match the statements/expressions given in Column-$$I$$ with the values given in Column-$$II.$$ $$\,\,\,\,\,$$ $$\,\,\,\,\,$$ $$\,\,\,\,\,$$ Column-$$I... Consider the following linear equations$$ax+by+cz=0;\,\,\,bx+cy+az=0;\,\,\,cx+ay+bz=0$$Match the conditions/expressions in Colu... Match the folowing : (A)$$\,\,\,$$Two rays$$x + y = \left| a \right|$$and$$ax - y=1$$intersects each other in the$$\,\,\,\,\,\,\,\,\,\,$$first q... Find the equation of the plane containing the line$$2x-y+z-3=0,3x+y+z=5$$and at a distance of$${1 \over {\sqrt 6 }}$$from the point$$(2, 1, -1).$...
If the incident ray on a surface is along the unit vector $$\widehat v\,\,,$$ the reflected ray is along the unit vector $$\widehat w\,\,$$ and the no...
A parallelopiped $$'S'$$ has base points $$A, B, C$$ and $$D$$ and upper face points $$A',$$ $$B',$$ $$C'$$ and $$D'.$$ This parallelopiped is compres...
Find the equation of plane passing through $$(1, 1, 1)$$ & parallel to the lines $${L_1},{L_2}$$ having direction ratios $$(1,0,-1),(1,-1,0).$$ Fi...
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ and $$\overrightarrow d$$ are distinct vectors such that $$\,\overrightarrow a \tim...$${P_1}$$and$${P_2}$$are planes passing through origin.$${L_1}$$and$${L_2}$$are two line on$${P_1}$$and$${P_2}$$respectively such that thei... If$$\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,$$are three non-coplanar unit vectors and$$\alpha ,\beta ,\gamma $$are the angles bet... (i) Find the equation of the plane passing through the points$$(2, 1, 0), (5, 0, 1)$$and$$(4, 1, 1).$$(ii) If$$P$$is the point$$(2, 1, 6)$$th... Let$$V$$be the volume of the parallelopiped formed by the vectors$$\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k,...
Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concur...
Find $$3-$$dimensional vectors $${\overrightarrow v _1},{\overrightarrow v _2},{\overrightarrow v _3}$$ satisfying $$\,{\overrightarrow v _1}.{\over... Let$$\overrightarrow A \left( t \right) = {f_1}\left( t \right)\widehat i + {f_2}\left( t \right)\widehat j$$and$$$\overrightarrow B \left( t \rig... Let $$u$$ and $$v$$ be units vectors. If $$w$$ is a vector such that $$w + \left( {w \times u} \right) = v,$$ then prove that $$\left| {\left( {u \tim... Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points ... For any two vectors$$u$$and$$v,$$prove that (a)$${\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v ... If $$A,B$$ and $$C$$ are vectors such that $$\left| B \right| = \left| C \right|.$$ Prove that $$\left[ {\left( {A + B} \right) \times \left( {A + C}... The position vectors of the vertices$$A, B$$and$$C$$of a tetrahedron$$ABCD$$are$$\widehat i + \widehat j + \widehat k,\,\widehat i$$and$$3\wi... If the vectors $$\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,$$ are not coplanar, then prove that the vector $$\left( {\overrightarrow a... In a triangle$$ABC, D$$and$$E$$are points on$$BC$$and$$AC$$respectively, such that$$BD=2DC$$and$$AE=3EC.$$Let$$P$$be the point of inters... Determine the value of$$'c'$$so that for all real$$x,$$the vector$$cx\widehat i - 6\widehat j - 3\widehat k$$and$$x\widehat i + 2\widehat j + ... Let $$\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrightarrow j + \overrightarrow k... If vectors$$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$are coplanar, show that$$$\left| {\matrix{ {} & {\overrightarrow {a.}...
In a triangle $$OAB,E$$ is the midpoint of $$BO$$ and $$D$$ is a point on $$AB$$ such that $$AD:DB=2:1.$$ If $$OD$$ and $$AE$$ intersect at $$P,$$ det...
Let $$OA$$ $$CB$$ be a parallelogram with $$O$$ at the origin and $$OC$$ a diagonal. Let $$D$$ be the midpoint of $$OA.$$ Using vector methods prove ...
If $$A, B, C, D$$ are any four points in space, prove that - $$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \ti... The position vectors of the points$$A, B, C$$and$$D$$are$$3\widehat i - 2\widehat j - \widehat k,\,2\widehat i + 3\widehat j - 4\widehat k,\, - \...
A vector $$\overrightarrow A$$ has components $${A_1},{A_2},{A_3}$$ in a right -handed rectangular Cartesian coordinate system $$oxyz.$$ The coordina...
$${A_1},{A_2},.................{A_n}$$ are the vertices of a regular plane polygon with $$n$$ sides and $$O$$ is its centre. Show that $$\sum\limits_... Find all values of$$\lambda $$such that$$x, y, z,\, \ne (0,0,0)$$and$$\left( {\overrightarrow i + \overrightarrow j + 3\overrightarrow...
From a point $$O$$ inside a triangle $$ABC,$$ perpendiculars $$OD$$, $$OE, OF$$ are drawn to the sides $$BC, CA, AB$$ respectively. Prove that the pe...

## Fill in the Blanks

Let $$OA=a,$$ $$OB=10a+2b$$ and $$OC=b$$ where $$O,A$$ and $$C$$ are non-collinear points. Let $$p$$ denote the area of the quadrilateral $$OABC,$$ an...
If $$\overrightarrow b \,$$ and $$\overrightarrow c \,$$ are two non-collinear unit vectors and $$\overrightarrow a \,$$ is any vector, then $$\left( ... A nonzero vector$$\overrightarrow a $$is parallel to the line of intersection of the plane determined by the vectors$$\widehat i,\widehat i + \wide...
A unit vector perpendicular to the plane determined by the points $$P\left( {1, - 1,2} \right)Q\left( {2,0, - 1} \right)$$ and $$R\left( {0,2,1} \righ... A unit vector coplanar with$$\overrightarrow i + \overrightarrow j + 2\overrightarrow k $$and$$\overrightarrow i + 2\overrightarrow j + \overri...
Given that $$\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = ... The components of a vector$$\overrightarrow a $$along and perpendicular to a non-zero vector$$\overrightarrow b $$are ......and .....respectively. If the vectors$$a\widehat i + \widehat j + \widehat k,\,\,\widehat i + b\widehat j + \widehat k$$and$$\widehat i + \widehat j + c\widehat k\...
Let $$b = 4\widehat i + 3\widehat j$$ and $$\overrightarrow c$$ be two vectors perpendicular to each other in the $$xy$$-plane. All vectors in the sa...
If $$\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr ... If$$\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C} $$are three non-coplannar vectors, then -$${{\overrightarrow A .\overrightarrow ...
If $$\overrightarrow A = \left( {1,1,1} \right),\,\,\overrightarrow C = \left( {0,1, - 1} \right)$$ are given vectors, then a vector $$B$$ satifying...
$$A, B, C$$ and $$D,$$ are four points in a plane with position vectors $$a, b, c$$ and $$d$$ respectively such that $$\left( {\overrightarrow a - \... The unit vector perpendicular to the plane determined by$$P\left( {1, - 1,2} \right),\,Q\left( {2,0, - 1} \right)$$and$$R\left( {0,2,1} \right)$$i... The area of the triangle whose vertices are$$A(1, -1, 2), B(2, 1, -1), C(3, -1, 2)$$is .......... Let$$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$be vectors of length$$3, 4, 5$$respectively. Let$$\overrightarrow A $$be perpendi... ## True or False For any three vectors$${\overrightarrow a ,\,\overrightarrow b ,}$$and$${\overrightarrow c ,}\left( {\overrightarrow a - \overrightarrow b }...
The points with position vectors $$a+b,$$ $$a-b,$$ and $$a+kb$$ are collinear for all real values of $$k.$$
If $$X.A=0, X.B=0, X.C=0$$ for some non-zero vector $$X,$$ then $$\left[ {A\,B\,C} \right] = 0$$
Let $$\overrightarrow A ,\overrightarrow B$$ and $${\overrightarrow C }$$ be unit vectors suppose that \overrightarrow A .\overrightarrow B = \ove...
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