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=‌=‌=−1‌yE=‌=‌=4‌zE=‌=‌=7So, the midpoint
E has coordinates
(−1,4,7).
Now, we can find the vector
by subtracting
B from
E :
=E−B=(−1−(−4),4−4,7−9)=(3,0,−2) The vector
is
(3,0,−2). To find the direction cosines, we need to normalize the vector. The length of vector
is given by:
||=√32+02+(−2)2=√9+0+4=√13The direction cosines are the normalized components of the vector
. Therefore, the direction cosines are:
(‌,‌‌‌,‌‌‌)This matches Option D:
‌,‌‌0,‌‌−‌⟩Thus, the correct answer is Option D.