Let $\vec{a}, \vec{b}, \vec{c}$ be non-coplanar vectors. If the three points with position vectors $\lambda \vec{a}-2 \vec{b}+\vec{c}$, $2 \vec{a}+\lambda \vec{b}-2 \vec{c}$, and $4 \vec{a}+7 \vec{b}-8 \vec{c}$ are collinear, then $\lambda=$

Let $D$ and $E$ be the midpoints of the sides $AC$ and $BC$ of a triangle $ABC$ respectively. If $O$ is an interior point of the triangle $ABC$ such that $\overrightarrow{OA}+2\overrightarrow{OB}+3\overrightarrow{OC}=\overrightarrow{0}$,then the area (in sq. units) of the triangle $ODE$ is

Let $v_1, v_2, v_3, v_4$ be unit vectors in the $XY$-plane,one each in the interior of the four quadrants. Which of the following statements is necessarily true?

If $|a| + |b| = |c|$ and $a + b = c$,then the angle between $a$ and $b$ is

The point of intersection of $r \times a = b \times a$ and $r \times b = a \times b$,where $a = i + j$ and $b = 2i - k$ is

If the position vectors of the vertices of a triangle are $2i + 4j - k,$ $4i + 5j + k,$ and $3i + 6j - 3k,$ then the triangle is