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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 351-400 of 401

  1. Match the statements given in Column I with the values given in Column II. Column I: (A) If $\vec a=\hat j+\sqrt3\hat k$, $\vec b=-\hat j+\sqrt3\hat k$ and …
  2. Match the entries in Column I with the entries in Column II. Column I: (A) In $\mathbb{R}^2$, if the magnitude of the projection vector of the vector …
  3. If vectors $\vec A,\vec B,\vec C$ are coplanar, show that $$\begin{vmatrix}\vec a & \vec b & \vec c\\ \vec a\cdot\vec a & \vec a\cdot\vec b & \vec a\cdot\vec c\\ \vec …
  4. If the vectors $\vec b,\vec c,\vec d$ are not coplanar, then prove that the vector …
  5. For any two vectors $u$ and $v$, prove that (a) $(u\cdot v)^{2}+|u\times v|^{2}=|u|^{2}|v|^{2}$ and (b) …
  6. Let $\vec u$ and $\vec v$ be unit vectors. If $\vec w$ is a vector such that $\vec w+(\vec w\times\vec u)=\vec v$, then prove that …
  7. If $\vec u,\vec v,\vec w$ are three non-coplanar unit vectors and $\alpha,\beta,\gamma$ are the angles between $\vec u$ and $\vec v$, $\vec v$ and $\vec w$, $\vec w$ and …
  8. Let $\hat a$ and $\hat b$ be two unit vectors. If the vectors $\vec c=\hat a+2\hat b$ and $\vec d=5\hat a-4\hat b$ are perpendicular to each other, then the angle …
  9. Let $ABCD$ be a parallelogram such that $\overrightarrow{AB}=\vec q$, $\overrightarrow{AD}=\vec p$ and $\angle BAD$ be an acute angle. If $\vec r$ is the vector that …
  10. $A_1,A_2,\ldots,A_n$ are the vertices of a regular plane polygon with $n$ sides and $O$ is its centre. Show that …
  11. Let $ABC$ and $PQR$ be any two triangles in the same plane. Assume that the perpendiculars from the points $A,B,C$ to the sides $QR,RP,PQ$ respectively are concurrent. …
  12. Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel …
  13. If $A$, $B$ and $C$ are vectors such that $|B|=|C|$. Prove that $[(A+B)\times(A+C)]\times(B\times C)\cdot(B+C)=0$.
  14. If $A,B,C,D$ are any four points in space, prove that …
  15. Let $V$ be the volume of the parallelopiped formed by the vectors $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$, $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$ …
  16. A parallelepiped $S$ has base points $A,B,C$ and $D$ and upper face points $A',B',C'$ and $D'$. This parallelepiped is compressed by the upper face $A'B'C'D'$ to form a …
  17. If $\vec a,\vec b,\vec c$ and $\vec d$ are distinct vectors such that $\vec a\times\vec c=\vec b\times\vec d$ and $\vec a\times\vec b=\vec c\times\vec d$, prove that …
  18. Let $\vec{a}=-\hat{i}+\hat{j}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$ be two given vectors. If a vector $\vec{c}$ satisfies $(\vec{a}\times\vec{c})+\vec{b}=\vec{0}$ and …
  19. Let a unit vector which makes an angle of $60^{\circ}$ with $2\hat{i}+2\hat{j}-\hat{k}$ and an angle of $45^{\circ}$ with $\hat{i}-\hat{k}$ be $\vec{C}$. Then …
  20. Let $ABC$ be a triangle of area $15\sqrt{2}$ and the vectors $\overrightarrow{AB}=\hat{i}+2\hat{j}-7\hat{k}$, $\overrightarrow{BC}=a\hat{i}+b\hat{j}+c\hat{k}$ and …
  21. For $\lambda>0$, let $\theta$ be the angle between the vectors $\vec{a}=\hat{i}+\lambda\hat{j}-3\hat{k}$ and $\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$. If the vectors …
  22. Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\vec{c}=x\hat{i}+2\hat{j}+3\hat{k}$, $x\in\mathbb{R}$. If $\vec{d}$ is the unit vector …
  23. Let $\vec{a}=\hat{i}-3\hat{j}+7\hat{k}$, $\vec{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that …
  24. If $A(1,-1,2)$, $B(5,7,-6)$, $C(3,4,-10)$ and $D(-1,-4,-2)$ are the vertices of a quadrilateral $ABCD$, then its area is :
  25. Let $\vec{a}=2\hat{i}+5\hat{j}-\hat{k}$, $\vec{b}=2\hat{i}-2\hat{j}+2\hat{k}$ and $\vec{c}$ be three vectors such that …
  26. Consider three vectors $\vec{a},\vec{b},\vec{c}$. Let $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{a}=\vec{b}\times\vec{c}$. If $\alpha\in\left[0,\dfrac{\pi}{3}\right]$ is the …
  27. Let the vectors $\vec{a},\vec{b},\vec{c}$ be such that $|\vec{a}|=2$, $|\vec{b}|=4$ and $|\vec{c}|=4$. If the projection of $\vec{b}$ on $\vec{a}$ is equal to the …
  28. If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $\sqrt{3}\left|\vec{a}+\vec{b}\right|+\left|\vec{a}-\vec{b}\right|$ is $\_\_\_\_\_\_\_\_$ .
  29. Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda\hat{i}+4\hat{k}$, $\lambda>0$, on the vector $\vec{a}=\hat{i}+2\hat{j}+2\hat{k}$. If $|\vec{a}+\vec{c}|=7$ …
  30. Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\dfrac{\pi}{3}$. If $\lambda\vec{a}+2\vec{b}$ and $3\vec{a}-\lambda\vec{b}$ are …
  31. Let the arc $AC$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $AC$, divides the arc $AC$ such that …
  32. Let the position vectors of three vertices of a triangle be $4\vec{p}+\vec{q}-3\vec{r}$, $-5\vec{p}+\vec{q}+2\vec{r}$ and $2\vec{p}-\vec{q}+2\vec{r}$. If the position …
  33. Let $\vec{a}=3\hat{i}-\hat{j}+2\hat{k}$, $\vec{b}=\vec{a}\times\left(\hat{i}-2\hat{k}\right)$ and $\vec{c}=\vec{b}\times\hat{k}$. Then the projection of …
  34. Let $A,B,C$ be three points in $xy$-plane, whose position vectors are given by $\sqrt{3}\hat{i}+\hat{j}$, $\hat{i}+\sqrt{3}\hat{j}$ and $a\hat{i}+(1-a)\hat{j}$ …
  35. If the components of $\vec{a}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$ along and perpendicular to $\vec{b}=3\hat{i}+\hat{j}-\hat{k}$ respectively, are …
  36. If $\vec a,\vec b,\vec c$ are vectors such that $[\vec a\ \vec b\ \vec c]=4$ then $[\vec a\times\vec b\ \ \vec b\times\vec c\ \ \vec c\times\vec a]=$
  37. If $\vec u,\vec v$ and $\vec w$ are three non-coplanar vectors, then $\left(\vec u+\vec v-\vec w\right)\cdot\left(\vec u-\vec v\right)\times\left(\vec v-\vec w\right)$ …
  38. If $\vec a,\vec b,\vec c$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\vec a+2\vec b+3\vec c$, $\lambda\vec b+4\vec c$ and …
  39. If $\vec a,\vec b,\vec c$ are non coplanar vectors and $\lambda$ is a real number, then …
  40. Let $\vec a=\hat i-\hat k$, $\vec b=x\hat i+\hat j+(1-x)\hat k$ and $\vec c=y\hat i+x\hat j+(1+x-y)\hat k$. Then $\left[\vec a,\vec b,\vec c\right]$ depends on :
  41. If $\left(\vec a\times\vec b\right)\times\vec c=\vec a\times\left(\vec b\times\vec c\right)$ where $\vec a,\vec b$ and $\vec c$ are any three vectors such that …
  42. Let $\vec a=\hat i+\hat j+\hat k$, $\vec b=\hat i-\hat j+2\hat k$ and $\vec c=x\hat i+(x-2)\hat j-\hat k$. If the vector $\vec c$ lies in the plane of $\vec a$ and …
  43. If $\vec u,\vec v,\vec w$ are non-coplanar vectors and $p,q$ are real numbers, then the equality …
  44. The vector $\vec a=\alpha\hat i+2\hat j+\beta\hat k$ lies in the plane of the vectors $\vec b=\hat i+\hat j$ and $\vec c=\hat j+\hat k$ and bisects the angle between …
  45. If $\vec a=\frac{1}{\sqrt{10}}\left(3\hat i+\hat k\right)$ and $\vec b=\frac17\left(2\hat i+3\hat j-6\hat k\right)$, then the value of …
  46. Let $\vec a=\hat j-\hat k$ and $\vec c=\hat i-\hat j-\hat k$. Then the vector $\vec b$ satisfying $\vec a\times\vec b+\vec c=\vec0$ and $\vec a\cdot\vec b=3$ is :
  47. If $\left[\vec a\times\vec b\ \ \vec b\times\vec c\ \ \vec c\times\vec a\right]=\lambda\left[\vec a\ \ \vec b\ \ \vec c\right]^{2}$ then $\lambda$ is equal to :
  48. Let $\vec a,\vec b$ and $\vec c$ be three non-zero vectors such that no two of them are collinear and …
  49. A vector $\vec a=\alpha\hat i+2\hat j+\beta\hat k$ $(\alpha,\beta\in\mathbb R)$ lies in the plane of the vectors, $\vec b=\hat i+\hat j$ and …
  50. For $p>0$, a vector $\vec{v}_2=2\hat{i}+(p+1)\hat{j}$ is obtained by rotating the vector $\vec{v}_1=\sqrt{3}p\,\hat{i}+\hat{j}$ by an angle $\theta$ about origin in …

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