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Yukti Vault

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 151-200 of 401

  1. For any vector $\vec a$, the value of $\left(\vec a\times\hat i\right)^{2}+\left(\vec a\times\hat j\right)^{2}+\left(\vec a\times\hat k\right)^{2}$ is equal to :
  2. $\vec a=3\hat i-5\hat j$ and $\vec b=6\hat i+3\hat j$ are two vectors and $\vec c$ is a vector such that $\vec c=\vec a\times\vec b$ then $|\vec a|:|\vec b|:|\vec c|=$
  3. Let $\vec u$ be a vector coplanar with the vectors $\vec a=2\hat i+3\hat j-\hat k$ and $\vec b=\hat j+\hat k$. If $\vec u$ is perpendicular to $\vec a$ and …
  4. Let $\vec a,\vec b$ and $\vec c$ be non-zero vectors such that $(\vec a\times\vec b)\times\vec c=\frac13|\vec b||\vec c|\vec a$. If $\theta$ is the acute angle between …
  5. If $|\vec a|=2$, $|\vec b|=3$ and $|2\vec a-\vec b|=5$, then $|2\vec a+\vec b|$ equals :
  6. The non-zero vectors $\vec a$, $\vec b$ and $\vec c$ are related by $\vec a=8\vec b$ and $\vec c=-7\vec b$. Then the angle between $\vec a$ and $\vec c$ is :
  7. A particle acted on by constant forces $4\hat i+\hat j-3\hat k$ and $3\hat i+\hat j-\hat k$ is displaced from the point $\hat i+2\hat j+3\hat k$ to the point …
  8. If $\vec a=\hat i+2\hat k$, $\vec b=\hat i+\hat j+\hat k$, $\vec c=7\hat i-3\hat j+4\hat k$, $\vec r\times\vec b+\vec b\times\vec c=\vec 0$ and $\vec r\cdot\vec a=0$ …
  9. Let $\vec a=-\hat i-\hat j+\hat k$, $\vec a\cdot\vec b=1$ and $\vec a\times\vec b=\hat i-\hat j$. Then $\vec a-6\vec b$ is equal to :
  10. If the four points, whose position vectors are $3\hat i-4\hat j+2\hat k$, $\hat i+2\hat j-\hat k$, $-2\hat i-\hat j+3\hat k$ and $5\hat i-2\alpha\hat j+4\hat k$ are …
  11. If $|\vec c|^2=60$ and $\vec c\times(\hat i+2\hat j+5\hat k)=\vec 0$, then a value of $\vec c\cdot(-7\hat i+2\hat j+3\hat k)$ is :
  12. Let $\vec a,\vec b,\vec c$ be three non-zero vectors which are pairwise non-collinear. If $\vec a+3\vec b$ is collinear with $\vec c$ and $\vec b+2\vec c$ is collinear …
  13. If the vector $\vec b=3\hat j+4\hat k$ is written as the sum of a vector $\vec b_1$, parallel to $\vec a=\hat i+\hat j$ and a vector $\vec b_2$, perpendicular to …
  14. The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8\hat i-6\hat j$ and $3\hat i+4\hat j-12\hat k$, is :
  15. Let $\vec a=2\hat i+\hat j-2\hat k$, $\vec b=\hat i+\hat j$. If $\vec c$ is a vector such that $\vec a\cdot\vec c=|\vec c|$, $|\vec c-\vec a|=2\sqrt2$ and the angle …
  16. The values of $a$, for which the points $A,B,C$ with position vectors $2\hat i-\hat j+\hat k$, $\hat i-3\hat j-5\hat k$ and $a\hat i-3\hat j+\hat k$ respectively are the …
  17. Let $\vec a,\vec b$ and $\vec c$ be three unit vectors such that $\vec a\times(\vec b\times\vec c)=\frac{\sqrt3}{2}(\vec b+\vec c)$. If $\vec b$ is not parallel to …
  18. Consider points $A,B,C$ and $D$ with position vectors $7\hat i-4\hat j+7\hat k$, $\hat i-6\hat j+10\hat k$, $-\hat i-3\hat j+4\hat k$ and $5\hat i-\hat j+5\hat k$ …
  19. Let $\vec a=\hat i+2\hat j+4\hat k$, $\vec b=\hat i+\lambda\hat j+4\hat k$ and $\vec c=2\hat i+4\hat j+\left(\lambda^{2}-1\right)\hat k$ be coplanar vectors. Then the …
  20. Let $\vec a=3\hat i+2\hat j+x\hat k$ and $\vec b=\hat i-\hat j+\hat k$, for some real $x$. Then $|\vec a\times\vec b|=r$ is possible if :
  21. Let $\vec a=4\hat i+3\hat j$ and $\vec b=3\hat i-4\hat j+5\hat k$ and $\vec c$ is a vector such that $\vec c\cdot(\vec a\times\vec b)+25=0$ …
  22. If $C$ is the mid point of $AB$ and $P$ is any point outside $AB$, then :
  23. The vector $\vec a=-\hat i+2\hat j+\hat k$ is rotated through a right angle, passing through the $y$-axis in its way and the resulting vector is $\vec b$. Then the …
  24. The vector equation of the plane through the line of intersection of the planes $x+y+z=1$ and $2x+3y+4z=5$ which is perpendicular to the plane $x-y+z=0$ is :
  25. The position vectors of the vertices $A$, $B$ and $C$ of a triangle are $2\hat i-3\hat j+3\hat k$, $2\hat i+2\hat j+3\hat k$ and $-\hat i+\hat j+3\hat k$ respectively. …
  26. The vectors $\overrightarrow{AB}=3\hat i+4\hat k$ & $\overrightarrow{AC}=5\hat i-2\hat j+4\hat k$ are the sides of triangle $ABC$. The length of the median through $A$ …
  27. Let $\vec u,\vec v,\vec w$ be such that $|\vec u|=1$, $|\vec v|=2$, $|\vec w|=3$. If the projection of $\vec v$ along $\vec u$ is equal to that of $\vec w$ along …
  28. Let $\vec a,\vec b$ and $\vec c$ be three non zero vectors such that $\vec b\cdot\vec c=0$ and $\vec a\times\left(\vec b\times\vec c\right)=\frac{\vec b-\vec c}{2}$. If …
  29. Let $\vec a,\vec b$ and $\vec c$ be three non-zero vectors such that no two of these are collinear. If the vector $\vec a+2\vec b$ is collinear with $\vec c$ and …
  30. If a unit vector $\vec a$ makes angles $\frac\pi3$ with $\hat i$, $\frac\pi4$ with $\hat j$ and $\theta\in(0,\pi)$ with $\hat k$, then a value of $\theta$ is :
  31. If $\hat u$ and $\hat v$ are unit vectors and $\theta$ is the acute angle between them, then $2\hat u\times3\hat v$ is a unit vector for :
  32. Let $\vec a=2\hat i+\lambda_1\hat j+3\hat k$, $\vec b=4\hat i+(3-\lambda_2)\hat j+6\hat k$ and $\vec c=3\hat i+6\hat j+(\lambda_3-1)\hat k$ be three vectors such that …
  33. Let $\vec a,\vec b$ and $\vec c$ be three unit vectors such that $\vec a+\vec b+\vec c=\vec 0$. If $\lambda=\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a$ and …
  34. Let the position vectors of the vertices $A$, $B$ and $C$ of a triangle be $2\hat i+2\hat j+\hat k$, $\hat i+2\hat j+2\hat k$ and $2\hat i+\hat j+2\hat k$ respectively. …
  35. Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors $A,B,C$ and $P$ are $\vec a,\vec b,\vec c$ and $\frac{\vec a+\vec b+\vec c}{4}$ …
  36. Let $\vec\alpha=(\lambda-2)\vec a+\vec b$ and $\vec\beta=(4\lambda-2)\vec a+3\vec b$ be two given vectors where vectors $\vec a$ and $\vec b$ are non-collinear. The …
  37. Let $\vec\alpha=3\hat i+\hat j$ and $\vec\beta=2\hat i-\hat j+3\hat k$. If $\vec\beta=\vec\beta_1-\vec\beta_2$, where $\vec\beta_1$ is parallel to $\vec\alpha$ and …
  38. If the vectors $\vec c$, $\vec a=x\hat i+y\hat j+z\hat k$ and $\vec b=\hat j$ are such that $\vec a,\vec c$ and $\vec b$ form a right handed system then $\vec c$ is :
  39. If $\vec x=3\hat i-6\hat j-\hat k$, $\vec y=\hat i+4\hat j-3\hat k$ and $\vec z=3\hat i-4\hat j-12\hat k$, then the magnitude of the projection of $\vec x\times\vec y$ …
  40. If the vectors $\vec a=\lambda\hat i+\mu\hat j+4\hat k$, $\vec b=-2\hat i+4\hat j-2\hat k$ and $\vec c=2\hat i+3\hat j+\hat k$ are coplanar and the projection of …
  41. If $\vec a,\vec b$ and $\vec c$ are unit vectors such that $\vec a+2\vec b+2\vec c=\vec 0$, then $|\vec a\times\vec c|$ is equal to
  42. If the position vectors of the vertices $A,B$ and $C$ of a $\triangle ABC$ are respectively $4\hat i+7\hat j+8\hat k$, $2\hat i+3\hat j+4\hat k$ and …
  43. Let a unit vector $\hat u=x\hat i+y\hat j+z\hat k$ make angles $\frac\pi2,\frac\pi3$ and $\frac{2\pi}{3}$ with the vectors $\frac1{\sqrt2}\hat i+\frac1{\sqrt2}\hat k$ …
  44. 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 $\vec a\cdot\vec c=4$, then …
  45. If the vectors $\vec a,\vec b$ and $\vec c$ form the sides $BC,CA$ and $AB$ respectively of a triangle $ABC$, then :
  46. Let $\overrightarrow{OA}=\vec a$, $\overrightarrow{OB}=12\vec a+4\vec b$ and $\overrightarrow{OC}=\vec b$, where $O$ is the origin. If $S$ is the parallelogram with …
  47. Let $P(3,2,3)$, $Q(4,6,2)$ and $R(7,3,2)$ be the vertices of $\Delta PQR$. Then, the angle $\angle QPR$ is
  48. If $|\vec a|=5$, $|\vec b|=4$, $|\vec c|=3$ thus what will be the value of $\left|\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a\right|$, given that …
  49. 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 …
  50. Let $\vec a=\hat i+\hat j+\sqrt2\hat k$, $\vec b=b_1\hat i+b_2\hat j+\sqrt2\hat k$ and $\vec c=5\hat i+\hat j+\sqrt2\hat k$ be three vectors such that the projection …

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