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Vectors & 3D

Vector Algebra: JEE Main & Advanced PYQs

401 previous year questions on Vector Algebra from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 301-350 of 401

  1. Let $\overrightarrow{AB}=2\hat i+4\hat j-5\hat k$ and $\overrightarrow{AD}=\hat i+2\hat j+\lambda\hat k$, $\lambda\in\mathbf{R}$. Let the projection of the vector …
  2. For a triangle $ABC$, let $\vec p=\overrightarrow{BC}$, $\vec q=\overrightarrow{CA}$ and $\vec r=\overrightarrow{BA}$. If $|\vec p|=2\sqrt3$, $|\vec q|=2$ and …
  3. Let $\vec a=-\hat i+2\hat j+2\hat k$, $\vec b=8\hat i+7\hat j-3\hat k$ and $\vec c$ be a vector such that $\vec a\times\vec c=\vec b$. If …
  4. Let $\vec a=\hat i-2\hat j+3\hat k$, $\vec b=2\hat i+\hat j-\hat k$, $\vec c=\lambda\hat i+\hat j+\hat k$ and $\vec v=\vec a\times\vec b$. If $\vec v\cdot\vec c=11$ and …
  5. Let $\vec a=-\hat i+\hat j+2\hat k$, $\vec b=\hat i-\hat j-3\hat k$, $\vec c=\vec a\times\vec b$ and $\vec d=\vec c\times\vec a$. Then $(\vec a-\vec b)\cdot\vec d$ is …
  6. Let $\vec a=2\hat i-5\hat j+5\hat k$ and $\vec b=\hat i-\hat j+3\hat k$. If $\vec c$ is a vector such that $2(\vec a\times\vec c)+3(\vec b\times\vec c)=\vec 0$ and …
  7. Let $\vec a=2\hat i+\hat j-2\hat k$, $\vec b=\hat i+\hat j$ and $\vec c=\vec a\times\vec b$. Let $\vec d$ be a vector such that $|\vec d-\vec a|=\sqrt{11}$ …
  8. For three unit vectors $\vec a,\vec b,\vec c$ satisfying $|\vec a-\vec b|^2+|\vec b-\vec c|^2+|\vec c-\vec a|^2=9$ and $|2\vec a+k\vec b+k\vec c|=3$, the positive value …
  9. Let $PQR$ be a triangle such that $\overrightarrow{PQ}=-2\hat i-\hat j+2\hat k$ and $\overrightarrow{PR}=a\hat i+b\hat j-4\hat k$, $a,b\in\mathbf{Z}$. Let $S$ be the …
  10. Let $P$ be a point in the plane of the vectors $\overrightarrow{AB}=3\hat i+\hat j-\hat k$ and $\overrightarrow{AC}=\hat i-\hat j+3\hat k$ such that $P$ is equidistant …
  11. Let the vectors $\overrightarrow{PQ}, \overrightarrow{QR}, \overrightarrow{RS}, \overrightarrow{ST}, \overrightarrow{TU}$ and $\overrightarrow{UP}$ represent the sides …
  12. The number of distinct real values of $\lambda$, for which the vectors $-\lambda^2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}-\lambda^2\hat{j}+\hat{k}$ and …
  13. Let the vectors $\vec a,\vec b,\vec c$ and $\vec d$ be such that $(\vec a\times\vec b)\times(\vec c\times\vec d)=\vec 0$. Let $P_1$ and $P_2$ be planes determined by the …
  14. Let $\vec{OA}=\vec a$, $\vec{OB}=10\vec a+2\vec b$ and $\vec{OC}=\vec b$ where $O$, $A$ and $C$ are non-collinear points. Let $p$ denote the area of the quadrilateral …
  15. State TRUE or FALSE: The points with position vectors $\vec a+\vec b$, $\vec a-\vec b$ and $\vec a+k\vec b$ are collinear for all real values of $k$.
  16. $A,B,C$ and $D$ are four points in a plane with position vectors $\vec a,\vec b,\vec c$ and $\vec d$ respectively such that …
  17. Let $\vec a,\vec b,\vec c$ be unit vectors such that $\vec a+\vec b+\vec c=\vec 0$. Which one of the following is correct?
  18. The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat a,\hat b,\hat c$ such that …
  19. Let two non-collinear unit vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow{OP}$ …
  20. Match List-I with List-II and select the correct answer using the code given below the lists: List I: P. Volume of parallelepiped determined by vectors $\vec a$ …
  21. Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi}{3}$. If …
  22. Let $\vec u$, $\vec v$ and $\vec w$ be vectors in three-dimensional space, where $\vec u$ and $\vec v$ are unit vectors which are not perpendicular to each other and …
  23. Let $\vec a=2\hat i+\hat j+\hat k$, and $\vec b$ and $\vec c$ be two nonzero vectors such that $|\vec a+\vec b+\vec c|=|\vec a+\vec b-\vec c|$ and $\vec b\cdot\vec c=0$. …
  24. For non-zero vectors $\vec a,\vec b,\vec c$, $\left|\left(\vec a\times\vec b\right)\cdot\vec c\right|=|\vec a|\,|\vec b|\,|\vec c|$ holds if and only if
  25. If $\vec a\times\vec b=\vec b\times\vec c=\vec c\times\vec a$ then $\vec a+\vec b+\vec c=$
  26. State TRUE or FALSE: If $\vec X\cdot\vec A=0$, $\vec X\cdot\vec B=0$, $\vec X\cdot\vec C=0$ for some non-zero vector $\vec X$, then …
  27. Let $\vec a_k=(\tan\theta_k)\hat i+\hat j$ and $\vec b_k=\hat i-(\cot\theta_k)\hat j$, where $\theta_k=\frac{2^{k-1}\pi}{2^{n}+1}$, for some $n\in\mathbb{N}$, $n>5$. …
  28. If $|\vec a|=4$, $|\vec b|=2$ and the angle between $\vec a$ and $\vec b$ is $\pi/6$ then $(\vec a\times\vec b)^{2}$ is equal to :
  29. Let $\vec w=\hat i+\hat j-2\hat k$, and $\vec u$ and $\vec v$ be two vectors, such that $\vec u\times\vec v=\vec w$ and $\vec v\times\vec w=\vec u$. Let …
  30. If $\vec a,\vec b,\vec c$ and $\vec d$ are unit vectors such that $$(\vec a\times\vec b)\cdot(\vec c\times\vec d)=1$$ and $\vec a\cdot\vec c=\frac12$, then
  31. Let P, Q, R and S be the points on the plane with position vectors $-2\hat i-\hat j$, $4\hat i$, $3\hat i+3\hat j$ and $-3\hat i+2\hat j$ respectively. The quadrilateral …
  32. If $\vec a$ and $\vec b$ are vectors in space given by $\vec a=\frac{\hat i-2\hat j}{\sqrt5}$ and $\vec b=\frac{2\hat i+\hat j+3\hat k}{\sqrt{14}}$, then the value of …
  33. Two adjacent sides of a parallelogram $ABCD$ are given by $\overrightarrow{AB}=2\hat i+10\hat j+11\hat k$ and $\overrightarrow{AD}=-\hat i+2\hat j+2\hat k$. The side …
  34. Let $\vec a=\hat i+\hat j+\hat k$, $\vec b=\hat i-\hat j+\hat k$ and $\vec c=\hat i-\hat j-\hat k$ be three vectors. A vector $\vec v$ in the plane of $\vec a$ and …
  35. Let $\vec a=-\hat i-\hat k$, $\vec b=-\hat i+\hat j$ and $\vec c=\hat i+2\hat j+3\hat k$ be three given vectors. If $\vec r$ is a vector such that …
  36. If $\vec a,\vec b$ and $\vec c$ are unit vectors satisfying $|\vec a-\vec b|^2+|\vec b-\vec c|^2+|\vec c-\vec a|^2=9$, then $|2\vec a+5\vec b+5\vec c|$ is
  37. If $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a} + \vec{b}| = \sqrt{29}$ and …
  38. Let $\overrightarrow{PR}=3\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{SQ}=\hat{i}-3\hat{j}-4\hat{k}$ determine diagonals of a parallelogram $PQRS$ and …
  39. Consider the set of eight vectors $V=\{a\hat{i}+b\hat{j}+c\hat{k}:a,b,c\in\{-1,1\}\}$. Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is
  40. Let $\Delta PQR$ be a triangle. Let $\vec a=\overrightarrow{QR}$, $\vec b=\overrightarrow{RP}$ and $\vec c=\overrightarrow{PQ}$. If $|\vec a|=12$, $|\vec b|=4\sqrt3$ and …
  41. Suppose that $\vec p$, $\vec q$ and $\vec r$ are three non-coplanar vectors in $\mathbb{R}^3$. Let the components of a vector $\vec s$ along $\vec p$, $\vec q$ and …
  42. Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point $S$ is such that $$\overrightarrow{OP}\cdot\overrightarrow{OQ} + …
  43. Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a}\cdot\vec{b}=0$. For some $x,y\in\mathbb{R}$, let $\vec{c}=x\vec{a}+y\vec{b}+(\vec{a}\times\vec{b})$. …
  44. Consider the cube in the first octant with sides $OP$, $OQ$ and $OR$ of length 1, along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O(0,0,0)$ is the …
  45. In a triangle $PQR$, let $\vec a=\overrightarrow{QR}$, $\vec b=\overrightarrow{RP}$ and $\vec c=\overrightarrow{PQ}$. If $$|\vec a|=3,\quad |\vec …
  46. Let the point $A$ divide the line segment joining the points $P(-1,-1,2)$ and $Q(5,5,10)$ internally in the ratio $r:1$ ($r>0$). If $O$ is the origin and …
  47. Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+2\hat{j}+\hat{k}$ and $\vec{d}=\vec{a}\times\vec{b}$. If $\vec{c}$ is a vector such that …
  48. $|\overrightarrow{OX}\times\overrightarrow{OY}|=$
  49. If the vectors $\vec a=\hat i-\hat j+2\hat k$, $\vec b=2\hat i+4\hat j+\hat k$ and $\vec c=\lambda\hat i+\hat j+\mu\hat k$ are mutually orthogonal, then $(\lambda,\mu)$ …
  50. If the vectors $\overrightarrow{AB}=3\hat i+4\hat k$ and $\overrightarrow{AC}=5\hat i-2\hat j+4\hat k$ are the sides of a triangle $ABC$, then the length of the median …

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