List of practice Questions

Let \(\overrightarrow a=\overrightarrow i+2\overrightarrow j+3 \overrightarrow k\) and \(\overrightarrow b= \overrightarrow i+ \overrightarrow j - \overrightarrow k\) . If \(\overrightarrow c\) is a vector such that \(\overrightarrow a. \overrightarrow c=11,\overrightarrow b.(\overrightarrow a.\overrightarrow c)=27\) and \(\overrightarrow b. \overrightarrow c=-\sqrt{3}|b|,\)  then \(|\overrightarrow a \times \overrightarrow c|^2\) is equal to _______ .

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 $-\hat{ i }+2 \hat{ j }-2 \hat{ k }$ is $30$, then $\alpha$ is equal to

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 \(\vec b×\vec c=\vec a\) and \(|\vec b|∈{1,2,…,10}\) is :

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 \(\vec{b}\) and \(\vec{c}\) are non-parallel, then the value of \(λ\) is

The angle $\theta$ between the vectors $\overrightarrow{a}=5\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}-\hat{k}$ is:

From the top of 30 m tower AB the angle of depression to another tower’s QP base and top is 60º and 30º respectively. Another point C lies on tower AB such that CQ is parallel to BP (where B and P are the base of towers). Then the area of BCQP is?