Answer:
[tex]8\hat i-2\hat j-2\hat k[/tex]
Explanation:
Vectors in 3D
Given a vector
[tex]\vec P = P_x\hat i+P_y\hat j+P_z\hat k[/tex]
A vector [tex]\vec Q[/tex] parallel to [tex]\vec P[/tex] is:
[tex]\vec Q = k.\vec P[/tex]
Where k is any constant different from zero.
We are given the vectors:
[tex]\vec A = \hat i+4\hat j-2\hat k[/tex]
[tex]\vec B = 3\hat i-5\hat j+\hat k[/tex]
It's not specified what the 'resultant' is about, we'll assume it's the result of the sum of both vectors, thus:
[tex]\vec A +\vec B = \hat i+4\hat j-2\hat k + 3\hat i-5\hat j+\hat k[/tex]
Adding each component separately:
[tex]\vec A +\vec B = 4\hat i-\hat j-\hat k[/tex]
To find a vector parallel to the sum, we select k=2:
[tex]2(\vec A +\vec B )= 8\hat i-2\hat j-2\hat k[/tex]
Thus one vector parallel to the resultant of both vectors is:
[tex]\mathbf{8\hat i-2\hat j-2\hat k}[/tex]
