JEE Main Vector Algebra PYQ (2002–2026)

5 real JEE Main questions on Vector Algebra, straight from past papers. Try them, then check the answer.

1. Q1JEE Main 2026 (Online) 24th January Morning Shift

   Let the lines $\mathrm{L}_1: \vec{r}$​${}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}), \lambda \in \mathbb{R}$ and $\mathrm{L}_2: \vec{r}$​${}=(4 \hat{\mathrm{i}}+\hat{\mathrm{j}})+\mu(5 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}), \mu \in \mathbb{R}$, intersect at the point R . Let P and Q be the points lying on lines $L_1$ and $L_2$, respectively, such that $|\overrightarrow{\mathrm{PR}}|$​${}=\sqrt{29}$ and $|\overrightarrow{\mathrm{PQ}}|$​${}=\sqrt{\frac{47}{3}}$. If the point P lies in the first octant, then $27(\mathrm{QR})^2$ is equal to

   1. A

      360

   2. B

      348

   3. C

      320

   4. D

      340

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2. Q2JEE Main 2025 (Online) 24th January Morning Shift

   Let $\vec{a}$​${}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}$​${}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{C}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c}$​${}=5$, then $|\vec{c}|$ is equal to

   1. A
      $\sqrt{\frac{11}{6}}$
   2. B
      $\frac{1}{3 \sqrt{2}}$
   3. C
      18
   4. D
      16
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3. Q3JEE Main 2024 (Online) 4th April Morning Shift

   Let ABC be a triangle of area $15 \sqrt{2}$ and the vectors $\overrightarrow{\mathrm{AB}}$​${}=\hat{i}+2 \hat{j}-7 \hat{k}, \overrightarrow{\mathrm{BC}}$​${}=\mathrm{a} \hat{i}+\mathrm{b} \hat{j}+\mathrm{c} \hat{k}$ and $\overrightarrow{\mathrm{AC}}$​${}=6 \hat{i}+\mathrm{d} \hat{j}-2 \hat{k}, \mathrm{~d}>0$. Then the square of the length of the largest side of the triangle ABC is _________.

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4. Q4JEE Main 2023 (Online) 6th April Morning Shift

   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 $-\hat{i}+5 \hat{j}+6 \hat{k}$. Let the set $S$​${}=\{\lambda \in \mathbb{R}$ : the points A, B, C and D are coplanar $\}$.
   
   Then $\sum\limits_{\lambda \in S}(\lambda+2)^{2}$ is equal to :

   1. A
      $\frac{37}{2}$
   2. B
      25
   3. C
      13
   4. D
      41
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5. Q5JEE Main 2022 (Online) 25th June Evening Shift

   Let $\overrightarrow b $​${}= \widehat i + \widehat j + \lambda \widehat k$, $\lambda$ $\in$ R. If $\overrightarrow a $ is a vector such that $\overrightarrow a \times \overrightarrow b $​${}= 13\widehat i - \widehat j - 4\widehat k$ and $\overrightarrow a \,.\,\overrightarrow b + 21 $​${}= 0$, then $\left( {\overrightarrow b - \overrightarrow a } \right).\,\left( {\widehat k - \widehat j} \right) + \left( {\overrightarrow b + \overrightarrow a } \right).\,\left( {\widehat i - \widehat k} \right)$ is equal to _____________.

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