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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 51-100 of 401

  1. Let $\hat a$ and $\hat b$ be two unit vectors such that the angle between them is $\frac\pi4$. If $\theta$ is the angle between the vectors $(\hat a+\hat b)$ and …
  2. Let three vectors $\vec{a}=\alpha\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=5\hat{i}+3\hat{j}+4\hat{k}$, $\vec{c}=x\hat{i}+y\hat{j}+z\hat{k}$ form a triangle such that …
  3. Let $ABCD$ be a tetrahedron such that the edges $AB$, $AC$ and $AD$ are mutually perpendicular. Let the areas of the triangles $ABC$, $ACD$ and $ADB$ be $5$, $6$ and $7$ …
  4. Let $\vec{a}=2\hat{i}+\alpha\hat{j}+\hat{k}$, $\vec{b}=-\hat{i}+\hat{k}$, $\vec{c}=\beta\hat{j}-\hat{k}$, where $\alpha$ and $\beta$ are integers and $\alpha\beta=-6$. …
  5. Let $\vec{a}$ and $\vec{b}$ be the vectors of the same magnitude such that $\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1$. …
  6. Let the angle $\theta$, $0<\theta<\frac{\pi}{2}$ between two unit vectors $\hat{a}$ and $\hat{b}$ be $\sin^{-1}\left(\frac{\sqrt{65}}{9}\right)$. If the vector …
  7. Let $\vec a,\vec b,\vec c$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is $14$ and …
  8. Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}$ and $\vec{c}=3\hat{i}-\hat{j}+\lambda\hat{k}$ be three vectors. Let $\vec{r}$ be a unit …
  9. Let the volume of a parallelopiped whose coterminous edges are given by $\vec u=\hat i+\hat j+\lambda\hat k$, $\vec v=\hat i+\hat j+3\hat k$ and …
  10. Let $\vec a=\hat i-2\hat j+\hat k$ and $\vec b=\hat i-\hat j+\hat k$ be two vectors. If $\vec c$ is a vector such that $\vec b\times\vec c=\vec b\times\vec a$ and …
  11. Let $\vec{a}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$ and $\vec{c}=\hat{i}-3\hat{j}+4\hat{k}$ be three vectors. If a vector $\vec{p}$ satisfies …
  12. Let $\hat{a},\hat{b}$ be unit vectors. If $\vec{c}$ be a vector such that the angle between $\hat{a}$ and $\vec{c}$ is $\frac{\pi}{12}$, and …
  13. Let $\vec a=\hat i-\hat j+2\hat k$ and let $\vec b$ be a vector such that $\vec a\times\vec b=2\hat i-\hat k$ and $\vec a\cdot\vec b=3$. Then the projection of $\vec b$ …
  14. Let $A,B,C$ be three points whose position vectors respectively are $\vec{a}=\hat{i}+4\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+\alpha\hat{j}+4\hat{k}$ …
  15. Between the following two statements: Statement I : Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying …
  16. Let $ABC$ be a triangle such that $\overrightarrow{BC}=\vec a$, $\overrightarrow{CA}=\vec b$, $\overrightarrow{AB}=\vec c$, $|\vec a|=6\sqrt2$, $|\vec b|=2\sqrt3$ and …
  17. Let $\overrightarrow{OA}=2\vec{a}$, $\overrightarrow{OB}=6\vec{a}+5\vec{b}$ and $\overrightarrow{OC}=3\vec{b}$, where $O$ is the origin. If the area of the parallelogram …
  18. Let $\vec{a}$ be a vector which is perpendicular to the vector $3\hat{i}+\frac{1}{2}\hat{j}+2\hat{k}$. If …
  19. Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, $a_i>0$, $i=1,2,3$ be a vector which makes equal angles with the coordinate axes $OX$, $OY$ and $OZ$. Also, let the …
  20. Let $\vec{a}=\alpha\hat{i}+3\hat{j}-\hat{k}$, $\vec{b}=3\hat{i}-\beta\hat{j}+4\hat{k}$ and $\vec{c}=\hat{i}+2\hat{j}-2\hat{k}$ where $\alpha,\beta\in\mathbf{R}$, be …
  21. Consider two vectors $\vec{u}=3\hat{i}-\hat{j}$ and $\vec{v}=2\hat{i}+\hat{j}-\lambda\hat{k}$, $\lambda>0$. The angle between them is given by …
  22. Let $S$ be the set of all $a\in\mathbb{R}$ for which the angle between the vectors $\vec u=a(\log_eb)\hat i-6\hat j+3\hat k$ and …
  23. Consider a $\triangle ABC$ where $A(1,3,2)$, $B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle BAC$ meets the line $BC$ at $D$, then the length of the …
  24. Let $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$, $\vec{b}=2\hat{i}-3\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$ be three given vectors. Let $\vec{v}$ be a vector in …
  25. Let $\vec{a}=4\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=11\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that …
  26. Let $\vec a=\alpha\hat i+\hat j-\hat k$ and $\vec b=2\hat i+\hat j-\alpha\hat k$, $\alpha>0$. If the projection of $\vec a\times\vec b$ on the vector …
  27. Let a vector $\vec a$ has magnitude $9$. Let a vector $\vec b$ be such that for every $(x,y)\in\mathbf{R}\times\mathbf{R}-\{(0,0)\}$, the vector $(x\vec a+y\vec b)$ is …
  28. Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be …
  29. If $\vec{a}$ is a nonzero vector such that its projections on the vectors $2\hat{i}-\hat{j}+2\hat{k}$, $\hat{i}+2\hat{j}-2\hat{k}$ and $\hat{k}$ are equal, then a unit …
  30. Let $\vec a=\hat i+\hat j+2\hat k$ and $\vec b=-\hat i+2\hat j+3\hat k$. Then the vector product …
  31. Let $\vec{a}=6\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $|\vec{c}|\ge6$, $\vec{a}\cdot\vec{c}=6|\vec{c}|$ …
  32. Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a}\times\hat{b})|=2$. If $\theta\in(0,\pi)$ is the angle between $\hat{a}$ and …
  33. Let $\vec{a}=2\hat{i}-3\hat{j}+\hat{k}$, $\vec{b}=3\hat{i}+2\hat{j}+5\hat{k}$ and a vector $\vec{c}$ be such that …
  34. Let $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+2\hat{k}$. Then the number of vectors $\vec{b}$ such that $\vec{b}\times\vec{c}=\vec{a}$ and …
  35. Let the vectors $\vec a=(1+t)\hat i+(1-t)\hat j+\hat k$, $\vec b=(1-t)\hat i+(1+t)\hat j+2\hat k$ and $\vec c=t\hat i-t\hat j+\hat k$, $t\in\mathbf{R}$ be such that for …
  36. Let $\vec{a}=-5\hat{i}+\hat{j}-3\hat{k}$, $\vec{b}=\hat{i}+2\hat{j}-4\hat{k}$ and …
  37. A vector $\vec a$ is parallel to the line of intersection of the plane determined by the vectors $\hat i,\hat i+\hat j$ and the plane determined by the vectors …
  38. Let $\vec{a}=2\hat{i}+\hat{j}-\hat{k}$, $\vec{b}=\left(\left(\vec{a}\times\left(\hat{i}+\hat{j}\right)\right)\times\hat{i}\right)\times\hat{i}$. Then the square of the …
  39. Let $\vec a=\alpha\hat i+\hat j+\beta\hat k$ and $\vec b=3\hat i-5\hat j+4\hat k$ be two vectors, such that $\vec a\times\vec b=-\hat i+9\hat i+12\hat k$. Then the …
  40. If $\vec{a}\cdot\vec{b}=1$, $\vec{b}\cdot\vec{c}=2$ and $\vec{c}\cdot\vec{a}=3$, then the value of …
  41. Let $\vec a=2\hat i-\hat j+5\hat k$ and $\vec b=\alpha\hat i+\beta\hat j+2\hat k$. If $((\vec a\times\vec b)\times\hat i)\cdot\hat k=\frac{23}{2}$, then …
  42. Let $\vec a=3\hat i+\hat j$ and $\vec b=\hat i+2\hat j+\hat k$. Let $\vec c$ be a vector satisfying $\vec a\times(\vec b\times\vec c)=\vec b+\lambda\vec c$. If $\vec b$ …
  43. Let $a,b,c\in\mathbf{R}$ be such that $a^2+b^2+c^2=1$. If $a\cos\theta=b\cos\left(\theta+\frac{2\pi}3\right)=c\cos\left(\theta+\frac{4\pi}3\right)$, where …
  44. Let a vector $\vec{c}$ be coplanar with the vectors $\vec{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. If the vector $\vec{c}$ also satisfies the …
  45. Let $\vec{a}$ and $\vec{b}$ be the vectors along the diagonals of a parallelogram having area $2\sqrt{2}$. Let the angle between $\vec{a}$ and $\vec{b}$ be acute …
  46. The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t\hat{i}+6\hat{j}-3\hat{k}$ and $\vec{b}=t\hat{i}-2\hat{j}-2\alpha t\hat{k}$ are inclined at an obtuse …
  47. Let $\vec{a}=\hat{i}+\alpha\hat{j}+\beta\hat{k}$, $\alpha,\beta\in\mathbb{R}$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is …
  48. Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b}\times\vec{a}|=2$. Then $\left|(\vec{b}\times\vec{a})-\vec{b}\right|^{2}$ is equal to
  49. If the volume of a parallelopiped, whose coterminus edges are given by the vectors $\vec a=\hat i+\hat j+n\hat k$, $\vec b=2\hat i+4\hat j-n\hat k$ and …
  50. Let $\vec{a}=9\hat{i}-13\hat{j}+25\hat{k}$, $\vec{b}=3\hat{i}+7\hat{j}-13\hat{k}$ and $\vec{c}=17\hat{i}-2\hat{j}+\hat{k}$ be three given vectors. If $\vec{r}$ is a …

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