Q 1 :

For three vectors $\overrightarrow{A}=(-x\hat{i}-6\hat{j}-2\hat{k}), \overrightarrow{B}=(-\hat{i}+4\hat{j}+3\hat{k})$ and $\overrightarrow{C}=(-8\hat{i}-\hat{j}+3\hat{k})$, if $\overrightarrow{A}·(\overrightarrow{B}\times \overrightarrow{C})=0$ 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 $\overrightarrow{A}=2\hat{i}+3n\hat{j}+2\hat{k}$ and $\overrightarrow{B}=2\hat{i}–2\hat{j}+4p\hat{k}$, respectively. If both the vectors are at right angle to each other, the value of $n^{–1}$ is __________.          [2025]

Q 3 :

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

• –1

   

• 2

   

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• 1

   

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