Question

The length of the projection of the line joining (1, 2, 3) and (-1, 4, 2) on the line which has direction ratios (2, 3, -6) is .

• A. $\frac{8}{7}$
• B. $\frac{7}{8}$
• C. $\frac{6}{5}$
• D. $\frac{5}{6}$

Answer 1

The correct option is A $\frac{8}{7}$

Projection of a vector a on another vector b is given by $\frac{a.b}{|b|}$
Now we need to find vectors a & b
We have seen how to find the direction ratios of a line joining two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ It is given by$(x_2-x_1,y_2-y_1,z_2-z_1)$
So we’ll have ,
Direction ratios of a = (-1 -1, 4 -2, 2 -3)
= (-2, 2 , -1)
And thus the vector a will be given by,
a = -2i + 2j - k
And , b = 2i + 3j - 6k
So, the projection of a on b =$\frac{(-2i+2j-k).(2i+3j-6k)}{|2i+3j-6k|}$ = $\frac{-4+6+6}{\sqrt{2^2+3^2+6^2}}$ (Using dot product of a and b)
=$\frac{8}{7}$
So, the projection of line segment joining (1, 2, 3) & (-1, 4, 2) on the line having direction ratios (2, 3, - 6) = $\frac{8}{7}$