Ans. Here and  

Since direction cosines of a line making angles  with the  and axes respectively are 

Therefore, the direction cosines of the required line are:

cos90∘=0;cos⁡90∘=0;cos135∘=−12√;cos⁡135∘=−12;cos45∘=12√cos⁡45∘=12

Ans. Let a line make equal angles  with the co-ordinate axes. 

Direction cosines of the line are ……….(i)

Putting  in eq. (i), direction cosines of the required line making equal angles with the co-ordinate axes are 

Direction cosines of a line making equal angles with the co-ordinate axes in the positive i.e., first octant are 

Ans. We know that if  are direction ratios of a line, then direction cosines of the line are: 

……….(i)

Here direction ratios of the line are 

Putting the values in eq. (i),

Hence, direction cosines of required line are .

Ans. The given points are A (2, 3, 4), B and C (5, 8, 7) 

Direction ratios of the line joining A and B are

 …….(i)

  =  (say)

Again Direction ratios of the line joining B and C are

 =  (say)……….(ii)

From eq. (i) and (ii),

Therefore, AB is parallel to BC. But point B is common to both AB and BC. Hence points A, B, C are collinear.

Ans. Direction ratios of the line joining A and B are  

Direction cosines of line AB are

Now Direction ratios of the line joining B and C are 

Direction cosines of line BC are

⇒−416+36+16√,−616+36+16√,−416+36+16√⇒−416+36+16,−616+36+16,−416+36+16

Direction ratios of the line joining C and A are 

Direction cosines of line CA are