Three Dimension...

A line passes through the points $$(6,-7,-1)$$ and $$(2,-3, 1)$$. If the angle a which the line makes with the positive direction of x-axis is acute, the direction cosines of the line are.

Prove that the points $$A=(1,2,3),B(3,4,7),C(-3,-2,-5)$$ are collinear & find the ratio in which $$B$$ divides $$AC$$. 

Equation of pair of lines passing through origin and making and angle $${\tan ^{ - 1}}2$$ with the lines $$4x-3y+7=0$$.

If $$\vec { a } ,\vec { b } ,\vec { c } $$ are three non-zero vectors, no two of which are collinear and the vector $$\vec { a } +\vec { b } $$ is collinear with $$\vec { c }, \vec { b } +\vec { c } $$ is collinear with $$\vec {a},$$ then $$\vec { a } +\vec { b } +\vec { c }$$ is equal to -

Find the equation of the plane if the foot of the perpendicular from origin to the plane is $$ (2, 3, -5 ) $$

If the points with position vectors $$10\hat { i } +\lambda \hat { j } ,3\hat { i } -\hat { j } $$ and $$4\hat { i } +5\hat { j } $$ are collinear then $$\lambda$$ is 

Point $$\left(\alpha,\beta,\gamma\right)$$ lies on the plane $$x+y+z=2$$. Let $$\overrightarrow{a}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$$ and $$\hat{k}\times \left(\hat{k}\times \overrightarrow{a}\right)=0$$ then $$\gamma=$$

If the points with position vectors $$60\hat{i}+3\hat{j}, 40\hat{i}-8\hat{j}$$ and $$a\hat{i}-52j$$ are collinear, then $$a=?$$

If $$A ( 2 \overline{i} - \overline{j} - 3 \overline{k} , B ( 4 \overline {i} + \overline{j} - \overline{k} )$$ and $$ D( \overline{i} - \overline{j} - 2 \overline{k})$$ then the vector equation of the plane parallel to $$ \overline{ABC} $$ and passing through the centroid of the tetrahedron $$ABCD$$ is :

The distance of the point $$3\hat{i}+5\hat{k}$$ from the line parallel to the vector $$6\hat{i}+\hat{j}-2\hat{k}$$ and passing through the point $$8\hat{i}+3\hat{j}+\hat{k}$$ is