Product of Two Vectors | Vectors jee mains previous year questions

Q 1 :

For three vectors $\vec{A} = (- x \hat{i} - 6 \hat{j} - 2 \hat{k}), \vec{B} = (- \hat{i} + 4 \hat{j} + 3 \hat{k})$ and $\vec{C} = (- 8 \hat{i} - \hat{j} + 3 \hat{k})$ , if $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$ then value of $x$ is _______ . [2024]

(4) $\vec{B} = - \hat{i} + 4 \hat{j} + 3 \hat{k} a n d \vec{C} = - 8 \hat{i} - \hat{j} + 3 \hat{k}$

$\vec{B} \times \vec{C} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 1 & 4 & 3 \\ - 8 & - 1 & 3 \end{matrix} |$ $= 15 \hat{i} - 21 \hat{j} + 33 \hat{k}$

$\vec{A} \cdot (\vec{B} \times \vec{C}) = (- x \hat{i} - 6 \hat{j} - 2 \hat{k}) \cdot (15 \hat{i} - 21 \hat{j} + 33 \hat{k})$

$0 = - 15 x + 126 - 66$

$\Rightarrow 15 x = 60 \Rightarrow x = 4$

Q 2 :

Two particles are located at equal distance from origin. The position vectors of those are represented by $\vec{A} = 2 \hat{i} + 3 n \hat{j} + 2 \hat{k}$ and $\vec{B} = 2 \hat{i} - 2 \hat{j} + 4 p \hat{k}$ , respectively. If both the vectors are at right angle to each other, the value of $n^{- 1}$ is __________. [2025]

(3)

$\vec{A} \cdot \vec{B} = 0 and$ $| \vec{A} |$ $=$ $| \vec{B} |$

$\Rightarrow$ 4 – 6n + 8p = 0

3n – 4p = 2 ... (i)

Also $4 + 9 n^{2} + 4 = 4 + 4 + 16 p^{2}$

$3 n = \pm 4 p$ ... (ii)

$\pm 4 p - 4 p = 2$

Taking –ve sign

–8p = 2

$p = - \frac{1}{4}, 3 n + 1 = 2 \Rightarrow n = \frac{1}{3}$

Q 3 :

If two vectors $\vec{P} = \hat{i} + 2 m \hat{j} + m \hat{k}$ and $\vec{Q} = 4 \hat{i} - 2 \hat{j} + m \hat{k}$ are perpendicular to each other. Then, the value of m will be [2023]

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(2)

$\vec{P} \cdot \vec{Q} = 0$

$(\hat{i} + 2 m \hat{j} + m \hat{k}) \cdot (4 \hat{i} - 2 \hat{j} + m \hat{k}) = 0 \Rightarrow 4 - 4 m + m^{2} = 0$

$\Rightarrow {(m - 2)}^{2} = 0 \Rightarrow m = 2$

Q 4 :

Vectors $a \hat{i} + b \hat{j} + \hat{k}$ and $2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ are perpendicular to each other when 3a + 2b = 7, the ratio of a to b is $\frac{x}{2}$ . The value of x is ________ . [2023]

(1)

For two perpendicular vectors

$(a \hat{i} + b \hat{j} + \hat{k}) \cdot (2 \hat{i} - 3 \hat{j} + 4 \hat{k}) = 0$

2a – 3b + 4 = 0

on solving, 2a – 3b = –4

also given 3a + 2b = 7

We get a = 1, b = 2

$\frac{a}{b} = \frac{x}{2} \Rightarrow x = \frac{2 a}{b} = \frac{2 \times 1}{2} = 1$

Q 5 :

If $\vec{P} = 3 \hat{i} + \sqrt{3} \hat{j} + 2 \hat{k}$ and $\vec{Q} = 4 \hat{i} + \sqrt{3} \hat{j} + 2.5 \hat{k}$ then, the unit vector in the direction of $\vec{P} \times \vec{Q}$ is $\frac{1}{x} (\sqrt{3} \hat{i} + \hat{j} - 2 \sqrt{3} \hat{k})$ . The value of 'x' is ________ . [2023]

(4)

$\bar{P} = 3 \hat{i} + \sqrt{3} \hat{j} + 2 \hat{k}$ , $\bar{Q} = 4 \hat{i} + \sqrt{3} \hat{j} + 2.5 \hat{k}$

$\bar{P} \times \bar{Q} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & \sqrt{3} & 2 \\ 4 & \sqrt{3} & 2.5 \end{matrix} |$

$= \hat{i} (2.5 \sqrt{3} - 2 \sqrt{3}) - \hat{j} (3 \times 2.5 - 8) + \hat{k} (3 \sqrt{3} - 4 \sqrt{3})$

$= 0.5 \sqrt{3} \hat{i} + 0.5 \hat{j} - \sqrt{3} \hat{k} = \frac{1}{2} (\sqrt{3} \hat{i} + \hat{j} - 2 \sqrt{3} \hat{k})$

$| \bar{P} \times \bar{Q} |$ $= \frac{1}{4} (\sqrt{3} \hat{i} + \hat{j} - 2 \sqrt{3} \hat{k})$

Q 6 :

When the position vector $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ changes sign as $- \vec{r}$ , which one of the following vector will not flip under sign change? [2026]

Linear momentum
Velocity
Acceleration
Angular momentum

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(4)

$\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$

$\vec{v} = \frac{d \vec{r}}{d t} = v_{x} \hat{i} + v_{y} \hat{j} + v_{z} \hat{k}$

$\vec{p} = m \vec{v}$

$\vec{L} = m (\vec{r} \times \vec{v}) = (x \hat{i} + y \hat{j} + z \hat{k}) \times m (v_{x} \hat{i} + v_{y} \hat{j} + v_{z} \hat{k})$

$When sign of \vec{r} changes, \vec{L} remains same.$

Topic Question Set

Q 1 :

For three vectors $\vec{A} = (- x \hat{i} - 6 \hat{j} - 2 \hat{k}), \vec{B} = (- \hat{i} + 4 \hat{j} + 3 \hat{k})$ and $\vec{C} = (- 8 \hat{i} - \hat{j} + 3 \hat{k})$ , if $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$ then value of $x$ is _______ . [2024]

Q 3 :

If two vectors $\vec{P} = \hat{i} + 2 m \hat{j} + m \hat{k}$ and $\vec{Q} = 4 \hat{i} - 2 \hat{j} + m \hat{k}$ are perpendicular to each other. Then, the value of m will be [2023]

Q 4 :

Vectors $a \hat{i} + b \hat{j} + \hat{k}$ and $2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ are perpendicular to each other when 3a + 2b = 7, the ratio of a to b is $\frac{x}{2}$ . The value of x is ________ . [2023]

Q 5 :

If $\vec{P} = 3 \hat{i} + \sqrt{3} \hat{j} + 2 \hat{k}$ and $\vec{Q} = 4 \hat{i} + \sqrt{3} \hat{j} + 2.5 \hat{k}$ then, the unit vector in the direction of $\vec{P} \times \vec{Q}$ is $\frac{1}{x} (\sqrt{3} \hat{i} + \hat{j} - 2 \sqrt{3} \hat{k})$ . The value of 'x' is ________ . [2023]

Q 6 :

When the position vector $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ changes sign as $- \vec{r}$ , which one of the following vector will not flip under sign change? [2026]

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Topic Question Set

Q 1 : For three vectors A→=(-xi^-6j^-2k^), B→=(-i^+4j^+3k^) and C→=(-8i^-j^+3k^), if A→·(B→×C→)=0 then value of x is _______ . [2024]

Q 2 : Two particles are located at equal distance from origin. The position vectors of those are represented by A→=2i^+3nj^+2k^ and B→=2i^–2j^+4pk^, respectively. If both the vectors are at right angle to each other, the value of n–1 is __________. [2025]

Q 3 : If two vectors P→=i^+2mj^+mk^ and Q→=4i^–2j^+mk^ are perpendicular to each other. Then, the value of m will be [2023]

Q 4 : Vectors ai^+bj^+k^ and 2i^–3j^+4k^ are perpendicular to each other when 3a + 2b = 7, the ratio of a to b is x2. The value of x is ________ . [2023]

Q 5 : If P→=3i^+3j^+2k^ and Q→=4i^+3j^+2.5k^ then, the unit vector in the direction of P→×Q→ is 1x(3i^+j^–23k^). The value of 'x' is ________ . [2023]

Q 6 : When the position vector r→=xi^+yj^+zk^ changes sign as -r→, which one of the following vector will not flip under sign change? [2026]

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Q 1 :

For three vectors $\vec{A} = (- x \hat{i} - 6 \hat{j} - 2 \hat{k}), \vec{B} = (- \hat{i} + 4 \hat{j} + 3 \hat{k})$ and $\vec{C} = (- 8 \hat{i} - \hat{j} + 3 \hat{k})$ , if $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$ then value of $x$ is _______ . [2024]

Q 3 :

If two vectors $\vec{P} = \hat{i} + 2 m \hat{j} + m \hat{k}$ and $\vec{Q} = 4 \hat{i} - 2 \hat{j} + m \hat{k}$ are perpendicular to each other. Then, the value of m will be [2023]

Q 4 :

Vectors $a \hat{i} + b \hat{j} + \hat{k}$ and $2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ are perpendicular to each other when 3a + 2b = 7, the ratio of a to b is $\frac{x}{2}$ . The value of x is ________ . [2023]

Q 5 :

If $\vec{P} = 3 \hat{i} + \sqrt{3} \hat{j} + 2 \hat{k}$ and $\vec{Q} = 4 \hat{i} + \sqrt{3} \hat{j} + 2.5 \hat{k}$ then, the unit vector in the direction of $\vec{P} \times \vec{Q}$ is $\frac{1}{x} (\sqrt{3} \hat{i} + \hat{j} - 2 \sqrt{3} \hat{k})$ . The value of 'x' is ________ . [2023]

Q 6 :

When the position vector $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ changes sign as $- \vec{r}$ , which one of the following vector will not flip under sign change? [2026]