For a triangle $ABC$ with vertices $A(1, 0, 0)$,$B(0, 1, 0)$,and $C(0, 0, 1)$,the angle $A = \dots$

If $|\vec{a}|=4$ and $|\vec{b}|=5$,then the values of $k$ for which $\vec{a}+k \vec{b}$ is perpendicular to $\vec{a}-k \vec{b}$ are

Let $O$ be the origin and $PQR$ be an arbitrary triangle. If a point $S$ satisfies the condition $\overrightarrow{OP} \cdot \overrightarrow{OQ} + \overrightarrow{OR} \cdot \overrightarrow{OS} = \overrightarrow{OR} \cdot \overrightarrow{OP} + \overrightarrow{OQ} \cdot \overrightarrow{OS} = \overrightarrow{OQ} \cdot \overrightarrow{OR} + \overrightarrow{OP} \cdot \overrightarrow{OS}$,then the point $S$ is the:

Let $\lambda \in \mathbb{Z}$,$\vec{a} = \lambda \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = 3 \hat{i} - \hat{j} + 2 \hat{k}$. Let $\vec{c}$ be a vector such that $(\vec{a} + \vec{b} + \vec{c}) \times \vec{c} = \vec{0}$,$\vec{a} \cdot \vec{c} = -17$ and $\vec{b} \cdot \vec{c} = -20$. Then $|\vec{c} \times (\lambda \hat{i} + \hat{j} + \hat{k})|^2$ is equal to

In triangle $ABC$,the point $P$ divides $BC$ internally in the ratio $3:4$ and $Q$ divides $CA$ internally in the ratio $5:3$. If $AP$ and $BQ$ intersect in a point $G$,then $G$ divides $AP$ internally in the ratio

Let the unit vectors $a$ and $b$ be perpendicular and the unit vector $c$ be inclined at an angle $\theta$ to both $a$ and $b$. If $c = \alpha a + \beta b + \gamma (a \times b)$,then