Respuesta :
Answer:
A) r•s = -21.99 units
B) rXs = 22.77 units (In the direction of k)
Completed question
Two vectors, r and s, lie in the xy plane. Their magnitudes are 4.25 and 7.45 units, respectively, and their directions are 312° and 86.0°, respectively, as measured counterclockwise from the positive x axis. What are the values of the following products?
A) r•s =
B) (r)X(s)=
Explanation:
A) dot product of r and s can be expressed mathematically as;
r•s = |r||s|cos(A) .......1
Where A is the angle between the two vectors.
|r| and |s| are the magnitude of vector r and s.
r•s = 4.25×7.45 ×cos(86-312) = -21.99 units
B) cross product of r and s can be expressed mathematically as;
rXs = |r||s|sin(A) .......2
Where A is the smallest angle between the two vectors
|r| and |s| are the magnitude of vector r and s.
A = 312-86 = 226
A = 360-226 = 134 smallest
rXs = 4.25 × 7.45 × sin134.
rXs = 22.77 units
In the direction of k
Answer:
(a) -22.00 units
(b) 22.77 units
Explanation:
Given vectors are r and s
Where;
r = |r| = 4.25 and ∠r = 312° measured anticlockwise
s = |s| = 7.45 and ∠s = 86° measured anticlockwise
First, let's calculate the angle between vectors r and s by representing them in the figure below;
y | /s
| /
| /
| /
|/ )86° x
|\) 48°
| \
| \
| \
| \ r
To get the acute angle between r and the +x axis, subtract the reflex angle of r (312°) from 360° as follows;
360 - 312 = 48°
As shown in the diagram, the angle between vectors r and s is 48° + 86° = 134°
Now,
(a) The (r)(s) represents the dot or scalar product of the two vectors and it is given as;
(r) (s) = r x s cos θ ---------------------------(i)
Where;
r = magnitude of vector r = 4.25
s = magnitude of vector s = 7.45
θ is the angle between the two vectors r and s = 134°
Substitute these values into equation (i) as follows;
(r) (s) = 4.25 x 7.45 cos 134°
(r) (s) = 4.25 x 7.45 x -0.6947
(r) (s) = 31.66 x -0.6947
(r) (s) = -22.00 units
(b) The (r) X (s) represents the vector product of the two vectors and it is given as;
(r) (s) = r x s sin θ ---------------------------(ii)
Where;
r = magnitude of vector r = 4.25
s = magnitude of vector s = 7.45
θ is the angle between the two vectors r and s = 134°
Substitute these values into equation (ii) as follows;
(r) (s) = 4.25 x 7.45 sin 134°
(r) (s) = 4.25 x 7.45 x 0.7193
(r) (s) = 31.66 x 0.7193
(r) (s) = 22.77 units
