Three Dimension...

$${ L }_{ 1 }$$ and $${ L }_{ 2 }$$ are two lines whose vector equations are $${ L }_{ 1 }:\vec { r } =\lambda \left[ \left( cos\theta +\sqrt { 3 }  \right) \hat { i } +\left( \sqrt { 2 } sin\theta  \right) \hat { j } +\left( cos\theta -\sqrt { 3 }  \right) \hat { k }  \right] $$
$${ L }_{ 2 }:\vec { r } =\mu \left( a\hat { i } +b\hat { j } +c\hat { k }  \right) ,$$ where$$\lambda$$ and $$\mu $$ are scalars and$$\alpha $$ is the acute angle between $${ L }_{ 1 }$$and$${ L }_{ 2 }$$ If the angle$$ '\alpha '$$ is independent of $$\theta $$ then the value of $$\alpha $$ is

Direction ratio of two lines are $$l_{1}, m_{1},n_{1}$$ and  $$l_{2},m_{2},n_{2}$$ then direction ratios of the line perpendicular to both the lines are

The plane passing through $$(1, 1, 1), (1, -1, 1)$$ and $$(-7, -3, -5)$$ is parallel to 

Each group from the alternatives represents lengths of sides of a triangleStare which group does not represent a right-angled triangle.

The lines $$\vec{r}=i-j+\lambda(2i+k)$$ and $$\vec{r}=(2i-j)+\mu(i+j-k)$$ intersect for

there are 20 points in the plane on three of which are collinear. the number of straight lines by joining them is

What is the area of the triangle with vertices $$(0,2,2),\,(2,0,-1)$$ and $$(3,4,0)$$ ?

In the three points with position vectors (1, a. b) : (a, b, 3) are collinear in space, then the value of a + b is 

The number of straight lines that can be drawn through any two points out of $$10$$ points, of which $$7$$ are collinear.

If the vectors $$2\hat{i} + 3\hat{j} , ~5\hat{i} + 6\hat{j} ,$$ and $$ 8\hat{i} +\lambda{\hat{j}}$$ have their initial points at $$(1 , 1)$$, then the value of $$\lambda$$ so that the vectors terminate on one straight line is