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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 101-150 of 401

  1. Let $\vec a=\hat i+\hat j+\hat k$, $\vec b$ and $\vec c=\hat j-\hat k$ be three vectors such that $\vec a\times\vec b=\vec c$ and $\vec a\cdot\vec b=1$. If the length of …
  2. The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha\hat{i}-2\hat{j}+2\hat{k}$ and $\alpha\hat{i}+2\alpha\hat{j}-2\hat{k}$ is …
  3. Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=3\hat{i}+2\hat{j}-\hat{k}$, $\vec{c}=\lambda\hat{j}+\mu\hat{k}$ and $\hat{d}$ be a unit vector such that …
  4. Let $\vec{p}=2\hat{i}+3\hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2\hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$ is …
  5. Let $\vec a=\hat i-\alpha\hat j+\beta\hat k$, $\vec b=3\hat i+\beta\hat j-\alpha\hat k$ and $\vec c=-\alpha\hat i-2\hat j+\hat k$, where $\alpha$ and $\beta$ are …
  6. If the vectors, $\vec p=(a+1)\hat i+a\hat j+a\hat k$, $\vec q=a\hat i+(a+1)\hat j+a\hat k$ and $\vec r=a\hat i+a\hat j+(a+1)\hat k$ ($a\in\mathbf{R}$) are coplanar and …
  7. Let $\vec{a}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+3\hat{k}$. If $\vec{b}$ is a vector such that $\vec{a}=\vec{b}\times\vec{c}$ and …
  8. Let $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$, $\vec{b}=3\hat{i}+4\hat{j}-5\hat{k}$ and a vector $\vec{c}$ be such that …
  9. Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$, $\vec{a}\cdot\vec{c}=7$ and $\vec{b}$ is perpendicular …
  10. Let $x_0$ be the point of local maxima of $f(x)=\vec a\cdot(\vec b\times\vec c)$, where $\vec a=x\hat i-2\hat j+3\hat k$, $\vec b=-2\hat i+x\hat j-\hat k$ and …
  11. Let $\vec{a}=2\hat{i}+\alpha\hat{j}+\hat{k}$ and $\vec{b}=\beta\hat{i}-5\hat{j}+\gamma\hat{k}$, where $\alpha,\beta$ and $\gamma$ are real numbers. If …
  12. Let $\vec{b}=\hat{i}+\hat{j}+\lambda\hat{k}$, $\lambda\in\mathbf{R}$. If $\vec{a}$ is a vector such that $\vec{a}\times\vec{b}=13\hat{i}-\hat{j}-4\hat{k}$ and …
  13. Let $\vec{a}=\hat{i}+2\hat{j}-\hat{k}$, $\vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that …
  14. Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$, $\vec{b}=3\hat{i}-3\hat{j}+3\hat{k}$, $\vec{c}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{d}$ be a vector such that …
  15. Let $\vec{c}$ be a vector perpendicular to the vectors $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$. If …
  16. If $\vec{a}=\alpha\hat{i}+\beta\hat{j}+3\hat{k}$, $\vec{b}=-\beta\hat{i}-\alpha\hat{j}-\hat{k}$ and $\vec{c}=\hat{i}-2\hat{j}-\hat{k}$ such that $\vec{a}\cdot\vec{b}=1$ …
  17. If $\vec{a}=2\hat{i}+\hat{j}+3\hat{k}$, $\vec{b}=3\hat{i}+3\hat{j}+\hat{k}$ and $\vec{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$ are coplanar vectors and …
  18. Let the three sides of a triangle $ABC$ be given by the vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$. Let $G$ be the …
  19. Let the position vectors of points `$A$' and `$B$' be $\hat i+\hat j+\hat k$ and $2\hat i+\hat j+3\hat k$, respectively. A point `$P$' divides the line segment $AB$ …
  20. The projection of the line segment joining the points $(1,-1,3)$ and $(2,-4,11)$ on the line joining the points $(-1,2,3)$ and $(3,-2,10)$ is ________.
  21. Let $\theta$ be the angle between the vectors $\vec{a}$ and $\vec{b}$, where $|\vec{a}|=4$, $|\vec{b}|=3$ and $\theta\in\left(\frac{\pi}{4},\frac{\pi}{3}\right)$. Then …
  22. Let $\vec{a}=3\hat{i}+2\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\vec{c}$ be a vector such that …
  23. If $\vec x$ and $\vec y$ be two non-zero vectors such that $|\vec x+\vec y|=|\vec x|$ and $2\vec x+\lambda\vec y$ is perpendicular to $\vec y$, then the value of …
  24. Let $\vec{a}=\hat{i}+5\hat{j}+\alpha\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}+\beta\hat{k}$ and $\vec{c}=-\hat{i}+2\hat{j}-3\hat{k}$ be three vectors such that …
  25. If $\vec a=2\hat i+\hat j+2\hat k$, then the value of …
  26. If $\left(\vec{a}+3\vec{b}\right)$ is perpendicular to $\left(7\vec{a}-5\vec{b}\right)$ and $\left(\vec{a}-4\vec{b}\right)$ is perpendicular to …
  27. Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot\vec{c}=11$ …
  28. Let $\vec a,\vec b$ and $\vec c$ be three unit vectors such that $|\vec a-\vec b|^2+|\vec a-\vec c|^2=8$. Then $|\vec a+2\vec b|^2+|\vec a+2\vec c|^2$ is equal to …
  29. Let $\vec a$ and $\vec b$ be two vectors such that $|\vec a+\vec b|^2=|\vec a|^2+2|\vec b|^2$, $\vec a\cdot\vec b=3$ and $|\vec a\times\vec b|^2=75$. Then $|\vec a|^2$ …
  30. Let $\vec{a}=\hat{i}+\alpha\hat{j}+3\hat{k}$ and $\vec{b}=3\hat{i}-\alpha\hat{j}+\hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the …
  31. If the projection of the vector $\hat{i}+2\hat{j}+\hat{k}$ on the sum of the two vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $-\lambda\hat{i}+2\hat{j}+3\hat{k}$ is $1$ …
  32. Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\vec{c}=4\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three vectors such that …
  33. Let $\vec{x}$ be a vector in the plane containing vectors $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. If the vector $\vec{x}$ is …
  34. Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1$, $|\vec{b}|=4$, and $\vec{a}\cdot\vec{b}=2$. If $\vec{c}=\left(2\vec{a}\times\vec{b}\right)-3\vec{b}$ …
  35. Let $\vec a,\vec b$ and $\vec c$ be three vectors such that $|\vec a|=\sqrt3$, $|\vec b|=5$, $\vec b\cdot\vec c=10$ and the angle between $\vec b$ and $\vec c$ is …
  36. Let $\vec{a}=6\hat{i}+9\hat{j}+12\hat{k}$, $\vec{b}=\alpha\hat{i}+11\hat{j}-2\hat{k}$ and $\vec{c}$ be vectors such that $\vec{a}\times\vec{c}=\vec{a}\times\vec{b}$. If …
  37. Let $OABC$ be a parallelogram, $O$ be the origin, $A(2,4,-5)$ and $C(b,2,3)$. If $P(a,a,a)$ and $Q(9-a^2,3,a-1)$, $a\in\mathbb{N}$, are two points such that the …
  38. If $\vec a$ and $\vec b$ are non-collinear vectors, then the value of $\alpha$ for which the vectors $\vec u=(\alpha-2)\vec a+\vec b$ and …
  39. The vector $\left(\hat i\times\vec a\cdot\vec b\right)\hat i+\left(\hat j\times\vec a\cdot\vec b\right)\hat j+\left(\hat k\times\vec a\cdot\vec b\right)\hat k$ is equal …
  40. Let $\vec a=2\hat i-\hat j+\hat k$, $\vec b=\hat i+2\hat j-\hat k$ and $\vec c=\hat i+\hat j-2\hat k$ be three vectors. A vector of the type $\vec b+\lambda\vec c$ for …
  41. If $\hat a$, $\hat b$ and $\hat c$ are unit vectors satisfying $\hat a-\sqrt3\hat b+\hat c=\vec 0$, then the angle between the vectors $\hat a$ and $\hat c$ is :
  42. Let $\sqrt3\hat i+\hat j$, $\hat i+\sqrt3\hat j$ and $\beta\hat i+(1-\beta)\hat j$ respectively be the position vectors of the points $A$, $B$ and $C$ with respect to …
  43. The vectors $\vec a$ and $\vec b$ are not perpendicular and $\vec c$ and $\vec d$ are two vectors satisfying $\vec b\times\vec c=\vec b\times\vec d$ and …
  44. $\vec a,\vec b,\vec c$ are $3$ vectors, such that $\vec a+\vec b+\vec c=\vec 0$, $|\vec a|=1$, $|\vec b|=2$, $|\vec c|=3$, then …
  45. Given $\vec a=2\hat i+\hat j-2\hat k$ and $\vec b=\hat i+\hat j$. Let $\vec c$ be a vector such that $|\vec c-\vec a|=3$, $|(\vec a\times\vec b)\times\vec c|=3$ and the …
  46. Let $a,b$ and $c$ be distinct non-negative numbers. If the vectors $a\hat i+a\hat j+c\hat k$, $\hat i+\hat k$ and $c\hat i+c\hat j+b\hat k$ lie in a plane, then $c$ is :
  47. If $\hat x,\hat y$ and $\hat z$ are three unit vectors in three dimensional space, then the minimum value of …
  48. Let $\vec a=2\hat i+\hat j-2\hat k$ and $\vec b=\hat i+\hat j$. If $\vec c$ is a vector such that $\vec a\cdot\vec c=|\vec c|$, $|\vec c-\vec a|=2\sqrt2$ and the angle …
  49. The magnitude of the projection of the vector $2\hat i+3\hat j+\hat k$ on the vector perpendicular to the plane containing the vectors $\hat i+\hat j+\hat k$ and …
  50. 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 $\vec u\cdot\hat n=0$ and …

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