COMEDK 2023 Shift 1 Paper

To determine the direction cosines of the median BE of the triangle ABC with vertices at A(−2,3,6),B(−4,4,9), and C(0,5,8), we first need to find the coordinates of point E, which is the midpoint of the side AC.
Using the midpoint formula, the coordinates of E are calculated as follows:
‌xE=‌

xA+xC

2

=‌

−2+0

2

=−1
‌yE=‌

yA+yC

2

=‌

3+5

2

=4
‌zE=‌

zA+zC

2

=‌

6+8

2

=7
So, the midpoint E has coordinates (−1,4,7).
Now, we can find the vector

→

BE

by subtracting B from E :

→

BE

=E−B=(−1−(−4),4−4,7−9)=(3,0,−2)
The vector

→

BE

is (3,0,−2). To find the direction cosines, we need to normalize the vector. The length of vector

→

BE

is given by:
|

→

BE

|=√32+02+(−2)2=√9+0+4=√13
The direction cosines are the normalized components of the vector

→

BE

. Therefore, the direction cosines are:
(‌

3

√13

,‌‌‌

0

√13

,‌‌‌

−2

√13

)
This matches Option D:
‌

3

√13

,‌‌0,‌‌−‌

2

√13

⟩
Thus, the correct answer is Option D.