Please help me with this its very hard to visualize the problem.
The analytical method of adding vectors expressed in terms of their components may be applied to vectors in three dimensions, for which graphical work is inconvenient. Find the magnitude of the resultant of the vectors A = l2*i- 37*j + 58*k and B = 5*i + 30*j -42*k, where i, j, and k are unit vectors along the x, y, and z axes, respectively.
just add all the i vectors, j vectors, and k vectors up respectively
Then magnitude is:
sqrt((i^2+j^2+k^2))
To find the magnitude of the resultant vector, we can use the formula for vector addition in three dimensions:
|R| = sqrt(Ax^2 + Ay^2 + Az^2)
Where Ax, Ay, and Az are the x, y, and z components of the vector A, respectively.
Let's calculate it step by step.
1. Given vectors A and B:
A = 2i - 37j + 58k
B = 5i + 30j - 42k
2. Determine the x, y, and z components of vector A and B:
Ax = 2, Ay = -37, Az = 58
Bx = 5, By = 30, Bz = -42
3. Calculate the square of the magnitude of the resultant vector:
|R|^2 = (Ax + Bx)^2 + (Ay + By)^2 + (Az + Bz)^2
= (2 + 5)^2 + (-37 + 30)^2 + (58 - 42)^2
4. Simplify:
= 7^2 + (-7)^2 + 16^2
= 49 + 49 + 256
= 354
5. Take the square root of the result to get the magnitude of the resultant vector:
|R| = sqrt(354)
|R| ≈ 18.81
Therefore, the magnitude of the resultant vector is approximately 18.81.