Vectors

The length of the perpendicular from a point $A(\vec{a})$ to the plane $\vec{r}\cdot\vec{n}=p$ is

$\overset{\hat{}}{i}\cdot(\overset{\hat{}}{j}\times\overset{\hat{}}{k})+\overset{\hat{}}{j}\cdot(\overset{¨}{k}\times\overset{\hat{}}{i})+\overset{\hat{}}{k}\cdot(\overset{\hat{}}{i}\times\overset{\hat{}}{j})=$

The vector equation of a line passing through the point having position vector $3\overset{\hat{}}{i}+\overset{\hat{}}{j}-\overset{\hat{}}{k}$ and parallel to the vector $2\overset{\hat{}}{i}-\overset{\hat{}}{j}+2\overset{\hat{}}{k}$ is

The projection of the vector $\overset{\hat{}}{i}-\overset{\hat{}}{j}$ on the vector $\overset{\hat{}}{i}+\overset{\hat{}}{j}$ is

If the vectors $3\overset{\hat{}}{i}-5\overset{\hat{}}{j}+\overset{\hat{}}{k}$ and $9\overset{\hat{}}{i}-15\overset{\hat{}}{j}+p\overset{\hat{}}{k}$ are collinear, then $p=$

If $\vec{a}=2\overset{\hat{}}{ı}-2\overset{\hat{}}{ȷ}+\overset{\hat{}}{k}$ and $\vec{c}=-\overset{\hat{}}{ı}+2\overset{\hat{}}{k}$ then $|\vec{c}|\vec{a}$ is equal to :

If $(\vec{a}+\vec{b})\cdot(\vec{a}+\vec{b})=|\vec{a}|^{2}+|\vec{b}|^{2},\vec{a}\neq 0,\vec{b}\neq 0$, then which of the following option is correct?

If $\bar{a}$ and $\bar{b}$ have the same magnitude and angle between them is $60^{\circ}$ and their scalar product is $\dfrac{1}{2}$, then $|\bar{a}|$ is $\_\_\_\_$

The value of $\theta \in\left(0, \dfrac{\pi}{2}\right)$ for which vectors $\bar{a}=(\sin \theta) \hat{i}+(\cos \theta) \hat{j}$ and $\bar{b}=\hat{i}-\sqrt{3} \hat{j}+2 \hat{k}$ are perpendicular is

If $\bar{a}=2 \hat{i}-\hat{j}+\hat{k}, \bar{b}=\hat{i}+\hat{j}-2 \hat{k}$ and $\bar{c}=4 \hat{i}-2 \hat{j}+\hat{k}$, then the unit vector in the direction of $3 \overline{a}+\overline{b}-2 \overline{c}$ is

If $a=\vec{a}=\overset{\hat{}}{i}+\overset{\hat{}}{j}-2\overset{\hat{}}{k},\vec{b}=2\overset{\hat{}}{i}-\overset{\hat{}}{j}+\overset{\hat{}}{k}$ and $\vec{c}=3\overset{\hat{}}{i}-\overset{\hat{}}{k}$ and $\vec{c}=m\vec{a}+n\vec{b}$, then $m+n$ is equal to

If $\bar{a}=4 \hat{i}+\hat{j}+\hat{k}^{\prime} \bar{b}=2 \hat{i}+\hat{j}+2 \hat{k}$ and $\bar{c}=3 \hat{i}+4 \hat{j}+5 \hat{k}$, then $(\bar{a}+\bar{b}) \cdot(\bar{b}+\bar{c})=$

Let $\vec{a}=\lambda \hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}+4 \hat{k}, \vec{c}=\hat{i}+\mu \hat{j}+\hat{k}$
If $\vec{a}$ is parallel to $\vec{b}$ and $\vec{b}$ is perpendicular to $\vec{c}$ then $\lambda-\mu=\ldots$

The value of $|\bar{a} \cdot \bar{b}|^{2}+|\bar{a} \times \bar{b}| \cdot|\bar{a} \times \bar{b}|$ is …

The volume of the parallelopiped whose coterminous edges are