Question

If a line passes through two points (1,2,3) & (4,5,6) then the direction cosines of that line would be -

• A. $\frac{2}{\sqrt{3}},\frac{2}{\sqrt{3}},\frac{2}{\sqrt{3}}$
• B. $\frac{1}{\sqrt{3}},\frac{2}{\sqrt{3}},\frac{1}{\sqrt{3}}$
• C. $\frac{2}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{2}{\sqrt{3}}$
• D. $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

Answer 1

The correct option is D $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

Let’s solve this question with the help of vectors. Let ‘P’ be the point having coordinates (1,2,3) and Q be the point having coordinates as (4,5,6).
So position vector OP ‘O’ being origin will be = 1i + 2j + 3k
And the position vector OQ will be = 4i + 5j + 6k
Thus the vector PQ can be given by (4i + 5j + 6k) - (1i + 2j + 3k) = 3i + 3j + 3k (Using parallelogram law)
This vector represents the line segment PQ.
We know that if a vector is ai + bj + ck then its direction ratios are a,b,c. Similarly here, the direction ratios will be 3, 3, 3.
Now we know the direction ratios so we can easily find the direction cosines, which will be -
$\frac{3}{\sqrt{(3{)}^2+(3{)}^2+(3{)}^2}},\frac{3}{\sqrt{(3{)}^2+(3{)}^2+(3{)}^2}},\frac{3}{\sqrt{(3{)}^2+(3{)}^2+(3{)}^2}}$
Or, $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$