Let a plane $P$ has the points $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. Let $L$ be the line through the point $A$ and parallel to the vector $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If the plane $P$ and line $L$ intersect at a point $B(0,3,2)$ and the distance from $A$ to $B$ is 3 units, then equations of the normal to the plane $P$ through $A$ are

Let $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{b}$ be two vectors such that $\mathbf{a} \cdot \mathbf{b}=1$, $\cos (\mathbf{a} \cdot \mathbf{b})=\frac{1}{3}$ and the components of $\mathbf{b}$ w.r.t. $(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}})$ be integers. Then, the number of possible vectors that represent $\mathbf{b}$ is

Let $\pi_1^{\prime}$ be the plane passing through the point $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and perpendicular to the vector $a \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\pi_2$ be the plane passing through the point $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and perpendicular to the vector $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$. If $\theta$ is the angle between the planes $\pi_1$ and $\pi_2$ and $\cos \theta=-\sqrt{\frac{3}{7}}$, then the integral value of $a$ is

If $\mathbf{a}$ and $\mathbf{b}$ are two vectors such that $\mathbf{a}=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+p \hat{\mathbf{k}}$, $|\mathbf{b}|=7, \mathbf{a} \cdot \mathbf{b}=4$ and $|\mathbf{a} \times \mathbf{b}|=5 \sqrt{17}$, then $p=$