Given the vectors = 2.00 + 8.00 and = 4.00 - 5.00, do the following.
(a) Draw the vector sum = + and the vector difference = - . (Do this on paper. Your instructor may ask you to turn the paper in.)
(b) Calculate and , first in terms of unit vectors.
Calculate and in terms of polar coordinates, with angles measured with respect to the +x axis.
vectors have direction. YOu have specified no directional vectors in your vectors.
Given the vectors A = 2.00i + 8.00j and B = 4.00i - 5.00j, do the following.
(a) Draw the vector sum C= A+ B and the vector difference D= A- B.
(b) Calculate C and D, first in terms of unit vectors.
Calculate C and D in terms of polar coordinates, with angles measured with respect to the +x axis.
To calculate the vector sum = + and vector difference = - , we can add or subtract the corresponding components of the two vectors.
(a) To draw the vector sum = + , we add the corresponding components:
= (2.00 + 4.00) + (8.00 - 5.00)
= 6.00 + 3.00
So, the vector sum is = 6.00 + 3.00.
Now, to draw the vector difference = - , we subtract the corresponding components:
= (2.00 - 4.00) + (8.00 - (-5.00))
= -2.00 + 13.00
So, the vector difference is = -2.00 + 13.00.
(b) To calculate the magnitudes of the vectors and , we can use the Pythagorean theorem:
Magnitude of = √((6.00)^2 + (3.00)^2)
Magnitude of = √(36.00 + 9.00)
Magnitude of = √45.00
Magnitude of ≈ 6.71
Similarly,
Magnitude of = √((-2.00)^2 + (13.00)^2)
Magnitude of = √(4.00 + 169.00)
Magnitude of = √173.00
Magnitude of ≈ 13.15
To calculate the direction of a vector in terms of polar coordinates, we can use the arctan function to find the angle with respect to the +x axis.
Direction of = tan^(-1)((3.00) / (6.00))
Direction of ≈ 26.57 degrees
Direction of = tan^(-1)((13.00) / (-2.00))
Direction of ≈ -82.89 degrees
So, the vector can be expressed in polar coordinates as magnitude = 6.71, angle = 26.57 degrees, and the vector can be expressed as magnitude = 13.15, angle = -82.89 degrees.