Three Dimension...

$$O$$ is the origin and $$A$$ is the point $$\displaystyle \left ( a,b,c \right ).$$ Find the direction cosines of the join of $$OA$$ and deduce the equation of the plane through $$A$$ at right angles to $$OA$$.

The direction cosines of the lines bisecting the internal angle $$\theta$$ between the lines whose direction cosines are $$l_{1},m_{1},n_{1}$$ and $$l_{2},m_{2},n_{2}$$ are

The angle between the line $$2x=3y=-z$$ and $$6x=-y=-4z$$ is

The projection of a directed line segment on the co-ordinate axes are $$12, 4, 3$$, the DC's of the line are

The projections of a line segment on $$x, y, z$$ axes are $$12, 4, 3$$. The length and the direction cosines of the line segments are

Equation of the plane through three points $$\displaystyle A,B,C$$ with position vectors $$\displaystyle -6\vec i+3\vec j+2\vec k, 3\vec i-2\vec j+4\vec k, 5\vec i+7\vec j+3\vec k$$ is

If the position vectors of the points $$A$$, $$B$$, and $$C$$ be $$i + j $$ , $$i - j$$ and $$ai + bj+ ck$$ respective;y , then the points $$A$$, $$B$$ and $$C$$ are collinear if:

The angle between the lines whose direction cosines are given by the equations $${l}^{2}+{m}^{2}-{n}^{2}=0,l+m+n=0$$ is

Find the unit vectors perpendicular to the following pair of vectors:

If direction cosines of two lines are proportional to $$(2,3,-6)$$ and $$(3,-4,5)$$, then the acute angle between them is