Given the displacement vectors A = (7.00 - 1.00 + 3.00 ) m and B = (6.00 + 5.00 - 3.00 ) m, find
the magnitudes of the following vectors and express each in terms of its rectangular components.
(a) C = A - B (b) D = 3A + B
To find the magnitude of a vector, you can use the formula:
Magnitude = √(x^2 + y^2 + z^2)
Where (x, y, z) are the rectangular components of the vector. Let's calculate the magnitudes of the given vectors:
(a) C = A - B
To find C, subtract the corresponding components of A and B:
C = (7.00 - 6.00) i + (-1.00 - 5.00) j + (3.00 - (-3.00)) k
= 1.00 i - 6.00 j + 6.00 k
Now, calculate the magnitude of C:
Magnitude of C = √(1.00^2 + (-6.00)^2 + 6.00^2)
= √(1.00 + 36.00 + 36.00)
= √73.00
So, the magnitude of C is √73.00.
(b) D = 3A + B
To find D, multiply A by 3 and add it to B:
D = (3 * 7.00) i + (3 * (-1.00)) j + (3 * 3.00) k + (6.00 i + 5.00 j - 3.00 k)
= 21.00 i - 3.00 j + 9.00 k + 6.00 i + 5.00 j - 3.00 k
= 27.00 i + 2.00 j + 6.00 k
Now, calculate the magnitude of D:
Magnitude of D = √(27.00^2 + 2.00^2 + 6.00^2)
= √(729.00 + 4.00 + 36.00)
= √769.00
So, the magnitude of D is √769.00.