If $a = i + j$ and $b = 2i - k$ are two vectors, then the point of intersection of two lines $r \times a = b \times a$ and $r \times b = a \times b$ is

If $a$ and $b$ are unlike vectors,then $a \cdot b =$

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a}$ is perpendicular to $\vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $120^\circ$. If $\vec{a} + \vec{c}$ is perpendicular to $\vec{b} + \vec{c}$, then:

If the coordinates of $A, B, C, D$ are $(2, 3, -1), (3, 5, -3), (1, 2, 3)$ and $(3, 5, 7)$ respectively,then what is the projection of $AB$ on $CD$?

Let $A = \hat{i} + 2 \hat{j}$. If $B$ is a vector in the $XY$ plane such that $(A + B) \cdot B = 15$ and $A \cdot B = 6$,then $|B|$ is

Show that the vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$ form the vertices of a right-angled triangle.