Vector A has a magnitude 12m and is angled at 60 degrees counterclockwise from the positive direction of the x axis of an xy coord. system. Also. Vector B = (12m)i + (8m)j on that same coord system. Rotate the system counterclockwise about the origin by 20 degrees to form an x'y' system. On this new system, what are (a) Vector A and (b) Vector B, both in unit-vector notation?
The rotation matrix, R(θ), for a rotation of θ counter-clockwise (CCW) is:
| cosθ -sinθ |
| sinθ cosθ |
Rotation of the basis by 20° CCW is the same as rotating the vectors 20° CW, or θ=-20°.
A = (12cos(60°),12sin(60°))
= (6 m, 6√3 m)
B = (12 m, 8 m)
Vector A in the new reference is therefore
A'
= R(-20°)A
=
| cos(-20°) -sin(-20°) | |6 |
| sin(-20°) cos(-20°) | |6√3|
=
(6cos(-20°)+6√3(-sin(-20°)), 6sin(-20°)+6√3 cos(-20°) )
= (5.64 + 3.55, -2.05+9.77)
= (9.19, 7.71)
(check: √(9.19²+7.71²)=12, OK)
The rotation of vector B' can be worked out similarly.
The new coordinate system has unit vectors i' and j'
Because of the 20 degree counterclockwise rotation,
i = cos 20 i' -sin 20 j'
j = cos 20 j' +sin 20 i'
For vector A,
A = cos 60 i + sin 60 j
Next, just make the substitution for i and j, and you get the same vector in trannsformed coordinates.
Do the same for Vector B
A =
To find the unit-vector notation of Vector A and Vector B in the new x'y' system after the counterclockwise rotation, follow these steps:
Step 1: Determine the new coordinates of the x-axis and y-axis in the x'y' system:
For a counterclockwise rotation of 20 degrees about the origin, the x-axis in the new system (x'y') remains the same as the x-axis in the original system (xy). The y-axis in the new system is obtained by rotating the positive y-axis of the original system counterclockwise by 20 degrees.
Step 2: Write the new unit vectors i' and j' in terms of i and j:
The unit vector i' in the x'y' system remains the same as the unit vector i in the original xy system, so i' = i.
To find j', we need to rotate the unit vector j in the original xy system counterclockwise by 20 degrees. Using a rotation matrix, we can find the new coordinates of j in terms of i and j:
j' = cos(20)j - sin(20)i
Step 3: Express Vector A and Vector B in unit-vector notation in the x'y' system:
(a) Vector A in the x'y' system:
To express Vector A in the x'y' system, we need to find its components along the new unit vectors i' and j'.
The magnitude of Vector A (12m) remains the same. The angle counterclockwise from the positive direction of the x-axis in the new x'y' system can be found by subtracting 20 degrees from the original angle of 60 degrees.
Using the cosine and sine identities, we can express Vector A in unit-vector notation in the x'y' system as:
Vector A' = Acos(60 - 20)i' + Asin(60 - 20)j'
where A' is the magnitude of Vector A in the x'y' system.
(b) Vector B in the x'y' system:
To express Vector B in the x'y' system, we simply need to write it in terms of the new unit vectors i' and j'.
Vector B' = (12m)i' + (8m)j'
Substituting the expressions for i' and j' from step 2, we get:
Vector B' = (12m)i' + (8m)(cos(20)j - sin(20)i)
Please note that these are the general formulas to express Vector A and Vector B in the x'y' system. To obtain the specific numerical values, you would need to evaluate the trigonometric functions and perform the necessary calculations using the given values.
To find the new representations of Vector A and Vector B in the x'y' system, we can follow these steps:
Step 1: Find the new angle of Vector A in the x'y' system:
Since the xy system is rotated counterclockwise by 20 degrees, the new angle for Vector A can be found by subtracting 20 degrees from the original angle of 60 degrees:
New angle = 60 degrees - 20 degrees = 40 degrees.
Step 2: Find the new representation of Vector A in unit-vector notation:
To represent Vector A in unit-vector notation, we need to find its x' and y' components. The x' component can be calculated using the magnitude of Vector A multiplied by the cosine of the new angle, and the y' component can be calculated using the magnitude of Vector A multiplied by the sine of the new angle.
Therefore:
x' = magnitude * cosine(angle) = 12m * cos(40 degrees)
y' = magnitude * sine(angle) = 12m * sin(40 degrees)
Step 3: Find the new representation of Vector B in unit-vector notation:
Vector B is already given in unit-vector notation as (12m)i + (8m)j. Since the rotation does not affect Vector B, its representation in the x'y' system remains the same.
Therefore, the answers are:
(a) The new representation of Vector A in unit-vector notation in the x'y' system is (12m * cos(40 degrees))i + (12m * sin(40 degrees))j.
(b) The representation of Vector B in the x'y' system remains (12m)i + (8m)j.