Addition of vectors:
vector u
has a magnitude of 20 and a direction of 0º with respect to the horizontal, vector v has a magnitude of 40 and a direction of 60º with respect to the horizontal.
a) Find the magnitude and direction of the resultant to the nearest whole
number.
Vector Sum:
The resultant of two vectors is simply the vector sum of the vectors. There are a handful of ways to present the resultant factor; the notation that shows the vector magnitude and direction is called the polar vector notation. An example of a vector presented in polar vector notation is
a∠θ where a is the magnitude and θis the angle that the vector makes with the horizontal axis.
Answer and Explanation:
Let's first present the vectors in rectangular vector notation.
For the vector →u of magnitude 20 and direction 0∘ to the horizontal axis, the vector is →u=^i20.
For the vector →v
of magnitude 40 and direction 60∘ to the horizontal axis, the vector is →v=^i40cos60∘+^j40sin60∘.
The resultant vector →w is the vector sum of the vectors, i.e.
→w=→u+→v
=^i20+^i40cos60∘+^j40sin60∘
=^i(20+40cos60∘)+^j(40sin60∘)
=^i40+^j20√3
For a vector ^ix+^jy, the magnitude of the vector is √x2+y2 and the direction above the horizontal axis is θ=tan−1(yx).
Let's use the formulas:
|→w|∠θ=√(40)2+(20√3)2∠tan−1(20√340)≈52.9∠40.9∘
The magnitude of the vector is about 53 units in the direction 41-degrees above the horizontal axis.
