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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 1-50 of 401

  1. If the points with position vectors $\alpha\hat{i}+10\hat{j}+13\hat{k}$, $6\hat{i}+11\hat{j}+11\hat{k}$, $\frac{9}{2}\hat{i}+\beta\hat{j}-8\hat{k}$ are collinear, then …
  2. Let a vector $\vec{a}$ be coplanar with vectors $\vec{b}=2\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$. If $\vec{a}$ is perpendicular to …
  3. If $\vec{a}$ and $\vec{b}$ are perpendicular, then $\vec{a}\times\left(\vec{a}\times\left(\vec{a}\times\left(\vec{a}\times\vec{b}\right)\right)\right)$ is equal to :
  4. Let $\vec{a}=4\hat{i}-\hat{j}+3\hat{k}$, $\vec{b}=10\hat{i}+2\hat{j}-\hat{k}$ and a vector $\vec{c}$ be such that …
  5. For any vector $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, with $10|a_i|<1$, $i=1,2,3$, consider the following statements : (A) : $\max\{|a_1|,|a_2|,|a_3|\}\le|\vec{a}|$ …
  6. In a triangle $ABC$, if $\left|\overrightarrow{BC}\right|=8$, $\left|\overrightarrow{CA}\right|=7$, $\left|\overrightarrow{AB}\right|=10$, then the projection of the …
  7. Let $\vec{a}=2\hat{i}+3\hat{j}+3\hat{k}$ and $\vec{b}=6\hat{i}+3\hat{j}+3\hat{k}$. Then the square of the area of the triangle with adjacent sides determined by the …
  8. Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j}$, $\hat{i}+\hat{k}$ and $\hat{i}-\hat{j}$ …
  9. If four distinct points with position vectors $\vec{a},\vec{b},\vec{c}$ and $\vec{d}$ are coplanar, then $[\vec{a}\ \vec{b}\ \vec{c}]$ is equal to
  10. If vectors $\vec{a}_1=x\hat{i}-\hat{j}+\hat{k}$ and $\vec{a}_2=\hat{i}+y\hat{j}+z\hat{k}$ are collinear, then a possible unit vector parallel to the vector …
  11. Let $\vec{a},\vec{b},\vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$ satisfies …
  12. Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2\hat{j}+3\hat{k}$. If the position vectors of the points $A$, $B$ and $C$ are …
  13. Let the vectors $\vec{u}_1=\hat{i}+\hat{j}+a\hat{k}$, $\vec{u}_2=\hat{i}+b\hat{j}+\hat{k}$ and $\vec{u}_3=c\hat{i}+\hat{j}+\hat{k}$ be coplanar. If the vectors …
  14. The sum of all values of $\alpha$, for which the points whose position vectors are $\hat{i}-2\hat{j}+3\hat{k}$, $2\hat{i}-3\hat{j}+4\hat{k}$ …
  15. Let $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}$, $\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$ and $\vec{c}=-\hat{i}+4\hat{j}+3\hat{k}$. If $\vec{d}$ is a vector perpendicular to both …
  16. If $|\vec{a}|=2$, $|\vec{b}|=5$ and $|\vec{a}\times\vec{b}|=8$, then $|\vec{a}\cdot\vec{b}|$ is equal to :
  17. An arc $PQ$ of a circle subtends a right angle at its centre $O$. The mid point of the arc $PQ$ is $R$. If $\overrightarrow{OP}=\vec{u}$, $\overrightarrow{OR}=\vec{v}$ …
  18. Let $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$ and $\vec{b}=7\hat{i}+\hat{j}-6\hat{k}$. If $\vec{r}\times\vec{a}=\vec{r}\times\vec{b}$ …
  19. Let $a,b$ and $c$ be distinct positive numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$, $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ are co-planar, then $c$ …
  20. Let the vectors $\vec{a},\vec{b},\vec{c}$ represent three coterminous edges of a parallelopiped of volume $V$. Then the volume of the parallelopiped, whose coterminous …
  21. Let $S$ be the set of all $(\lambda,\mu)$ for which the vectors $\lambda\hat{i}-\hat{j}+\hat{k}$, $\hat{i}+2\hat{j}+\mu\hat{k}$ and $3\hat{i}-4\hat{j}+5\hat{k}$, where …
  22. Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1$, $\vec{a}\cdot\vec{b}=2$ and …
  23. Let for a triangle $ABC$, $\overrightarrow{AB}=-2\hat{i}+\hat{j}+3\hat{k}$, $\overrightarrow{CB}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$ …
  24. Let $\lambda\in\mathbb{Z}$, $\vec{a}=\lambda\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{c}$ be a vector such that …
  25. Let $ABCD$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and …
  26. Let the vectors $(2+a+b)\hat{i}+(a+2b+c)\hat{j}-(b+c)\hat{k}$, $(1+b)\hat{i}+2b\hat{j}-b\hat{k}$ and $(2+b)\hat{i}+2b\hat{j}+(1-b)\hat{k}$, $a,b,c\in\mathbf{R}$ be …
  27. A vector $\vec{a}$ has components $3p$ and $1$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the …
  28. Let $a,b,c$ be three distinct real numbers, none equal to one. If the vectors $a\hat{i}+\hat{j}+\hat{k}$, $\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are …
  29. Let $O$ be the origin, $\overrightarrow{OP}=\vec{a}$ and $\overrightarrow{OQ}=\vec{b}$. If $R$ is the point on $\overrightarrow{OP}$ such that …
  30. Let $\vec{a}=2\hat{i}+7\hat{j}-\hat{k}$, $\vec{b}=3\hat{i}+5\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both …
  31. Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$, $\vec{b}=3\left(\hat{i}-\hat{j}+\hat{k}\right)$. Let $\vec{c}$ be the vector such that $\vec{a}\times\vec{c}=\vec{b}$ and …
  32. Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|2\vec{a}+3\vec{b}|=|3\vec{a}+\vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ}$. If …
  33. Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $|\vec{a}|=|\vec{b}|$. If $|\vec{a}\times\vec{b}|=|\vec{a}|$, then the angle between …
  34. Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{b}=2\hat{i}-3\hat{j}+5\hat{k}$. If $\vec{r}\times\vec{a}=\vec{b}\times\vec{r}$ …
  35. Let $O$ be the origin. Let $\overrightarrow{OP}=x\hat{i}+y\hat{j}-\hat{k}$ and $\overrightarrow{OQ}=-\hat{i}+2\hat{j}+3x\hat{k}$, $x,y\in\mathbf{R}$, $x>0$, be such that …
  36. Let the position vectors of the points $A,B,C$ and $D$ be $5\hat{i}+5\hat{j}+2\lambda\hat{k}$, $\hat{i}+2\hat{j}+3\hat{k}$, $-2\hat{i}+\lambda\hat{j}+4\hat{k}$ and …
  37. If the points $P$ and $Q$ are respectively the circumcenter and the orthocentre of a $\triangle ABC$, then $\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}$ …
  38. Let $|\vec{a}|=2$, $|\vec{b}|=3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{\pi}{4}$. Then …
  39. Two adjacent sides of a parallelogram $PQRS$ are given by $\overrightarrow{PQ}=\hat{j}+\hat{k}$ and $\overrightarrow{PS}=\hat{i}-\hat{j}$. If the side $PS$ is rotated …
  40. Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $ABCD$. If the diagonal $\overrightarrow{BD}=\hat{i}+2\hat{j}+3\hat{k}$, then the area of the …
  41. Let $\vec{a}=\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$. If a vector $\vec{d}$ satisfies …
  42. Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5\vec{b}$ is collinear with $\vec{c}$ …
  43. Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a}\times\vec{c}=\vec{b}$ and $\vec{a}\cdot\vec{c}=3$, then …
  44. Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{a}\times\vec{b}=\vec{c}$, $\vec{b}\times\vec{c}=\vec{a}$ and $|\vec{a}|=2$. Then which one of the …
  45. Let $\hat{u}$ and $\hat{v}$ be unit vectors inclined at an acute angle such that $|\hat{u}\times\hat{v}|=\frac{\sqrt{3}}{2}$. If …
  46. Let $O$ be the origin and the position vectors of $A$ and $B$ be $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ respectively. If the internal bisector of …
  47. Let the vectors $\vec{a}=-\hat{i}+\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}+\hat{k}$. For some $\lambda,\mu\in\mathbf{R}$, let $\vec{c}=\lambda\vec{a}+\mu\vec{b}$. …
  48. If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=4\hat{i}-2\hat{j}+3\hat{k}$ and $\vec{c}=\hat{i}-2\hat{j}+\hat{k}$ are three vectors, then a vector of magnitude $6$ …
  49. Let $\vec a,\vec b$ and $\vec c$ be three vectors such that $\vec a=\vec b\times(\vec b\times\vec c)$. If magnitudes of the vectors $\vec a,\vec b$ and $\vec c$ are …
  50. Let $\vec{a}=\alpha\hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=-2\hat{i}+\alpha\hat{j}+\hat{k}$, where $\alpha\in\mathbf{R}$. If the area of the parallelogram whose adjacent …

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