Explanation:
Direction Cosine (l) = a_x / |a|
Direction Cosine (m) = a_y / |a|
Direction Cosine (n) = a_z / |a|
Where:
(a_x, a_y, a_z) are the components of the vector.
|a| is the magnitude of the vector.
In your case, you have the vector a = i + j - 2k, and its components are:
a_x = 1
a_y = 1
a_z = -2
To find the magnitude of the vector |a|, you can use the formula:
|a| = sqrt(a_x^2 + a_y^2 + a_z^2)
|a| = sqrt(1^2 + 1^2 + (-2)^2)
|a| = sqrt(1 + 1 + 4)
|a| = sqrt(6)
Now, you can find the direction cosines:
Direction Cosine (l) = a_x / |a| = 1 / sqrt(6)
Direction Cosine (m) = a_y / |a| = 1 / sqrt(6)
Direction Cosine (n) = a_z / |a| = -2 / sqrt(6)
So, the direction cosines of the vector a = i + j - 2k are:
l = 1 / sqrt(6)
m = 1 / sqrt(6)
n = -2 / sqrt(6)