Direction Cosines And Direction Ratios Of A Line

We are given the vertices of the triangle

A(-2%2C%203%2C%206)

B(-4%2C%204%2C%209)

, and

C(0%2C%205%2C%208)

, and we are asked to find the direction cosines of the median

BE

, where

E

is the midpoint of the side

AC

. Step 1: Find the coordinates of the midpoint

E

AC

. The midpoint formula in three dimensions is:

E%20%3D%20%5Cleft(%20%5Cfrac%7Bx_A%20%2B%20x_C%7D%7B2%7D%2C%20%5Cfrac%7By_A%20%2B%20y_C%7D%7B2%7D%2C%20%5Cfrac%7Bz_A%20%2B%20z_C%7D%7B2%7D%20%5Cright)

Substituting the coordinates of

A(-2%2C%203%2C%206)

and

C(0%2C%205%2C%208)

E%20%3D%20%5Cleft(%20%5Cfrac%7B-2%20%2B%200%7D%7B2%7D%2C%20%5Cfrac%7B3%20%2B%205%7D%7B2%7D%2C%20%5Cfrac%7B6%20%2B%208%7D%7B2%7D%20%5Cright)%20%3D%20%5Cleft(%20-1%2C%204%2C%207%20%5Cright)

Step 2: Find the direction ratios of the median

BE

. The direction ratios of the median

BE

are the differences between the coordinates of

B(-4%2C%204%2C%209)

and

E(-1%2C%204%2C%207)

%5Ctext%7BDirection%20ratios%20of%20%7D%20BE%20%3D%20(x_B%20-%20x_E%2C%20y_B%20-%20y_E%2C%20z_B%20-%20z_E)%20%3D%20(-4%20-%20(-1)%2C%204%20-%204%2C%209%20-%207)%20%3D%20(-3%2C%200%2C%202)

Step 3: Find the direction cosines of

BE

. The direction cosines are the normalized direction ratios. The magnitude of the direction ratios is:

%5Ctext%7BMagnitude%7D%20%3D%20%5Csqrt%7B(-3)%5E2%20%2B%200%5E2%20%2B%202%5E2%7D%20%3D%20%5Csqrt%7B9%20%2B%204%7D%20%3D%20%5Csqrt%7B13%7D

Thus, the direction cosines of

BE

are:

%5Cleft%5Clangle%20%5Cfrac%7B-3%7D%7B%5Csqrt%7B13%7D%7D%2C%20%5Cfrac%7B0%7D%7B%5Csqrt%7B13%7D%7D%2C%20%5Cfrac%7B2%7D%7B%5Csqrt%7B13%7D%7D%20%5Cright%5Crangle%20%3D%20%5Cleft%5Clangle%20%5Cfrac%7B3%7D%7B%5Csqrt%7B13%7D%7D%2C%200%2C%20-%5Cfrac%7B2%7D%7B%5Csqrt%7B13%7D%7D%20%5Cright%5Crangle

Therefore, the correct answer is option (D)

Chapter Questions
1 MCQs