(a) Express the vectors A, B, and C in the figure below in terms of unit vectors.
(b) Use unit vector notation to find the vectors R = A + B + C and S = C – A – B.
(c) What are the magnitude and directions of vectors R and S?
Well the picture goes and has a vector
A pointing in the 1st quadrant of the graph and between the y axis it has a angle of 37 deg
and a magnitude of 12m
vector B is in the 4th quadrant and the angle from the x axis to the vector is 40 deg
and a magnitude of 15m
vector C is in the 3rd quadrant and measured from the x axis it's angle is 60 deg
and a magnitude of 6.0m
~What are unit vectors? Would that just refer to the m? And does that include the i, j, and k hat? (i [^ on top of i])
How would I find these? like for C it has the angle of 60 deg from the x axis in the 3rd quadrant but if I was finding the x component of C I wouldn't use 60 right for the angle but rather 180+60 = 240 deg.
I'm not sure if that's correct but that has gotten me confused with unit vector.
And for the B vector in the 4th quadrant wouldn't I find the x and y component using not 40 deg but rather -40 deg?
~Thanks~
I really need to solve this by today...
From your description, and lettng i and j be unit vectors along the +x and +y axes,
A = 12.00 (sin 37 i + cos 37 j)
= 6.00 i + 9.58 j
B = 15.00 (cos 40 i - sin 40 j)
= 11.49 i - 9.64 j
C = 5.00 (-cos 60 i -sin 60 j)
= -2.50 i - 4.33 j
The changes of sign here and there, and the switch from sin to cos for i, are due to your changing definition of the quadrant the vector is in and the changing reference axis (y for A, +x for B and -x for C).
Now that you have defined A, B, and C with unit vectors, the calculation of R = A+B+C and S = C-A-B is easy. Once you have R & S in unit vector notation, switch back to magnitude/angle notation
Why is
A = 12.00 (sin 37 i + cos 37 j)
= 6.00 i + 9.58 j ??
I got when I plugged it into my calculator in degrees mode
7.22 i + 9.58 j
Well for
R= A+ B + C
R= (7.22i + 9.58j) + (11.49i- 9.64j) + (-2.50i - 4.33j) = (16.21i - 4.39j)m
S= C-A-B
S= (-2.50i - 4.33j)- (7.22i + 9.58j) - (11.49i - 9.64j)=
-2.50i - 4.33j- 7.22i - 9.58j -11.49i + 9.64j= (-21.21i- 4.27j)m
For step
c.) I don't know how to switch back to magnitude/ angle notation...does that include using tan ?
Um PLEASE help me out on this part c..seriously ...I did show what I did for a and b and I just need to know how c would work...
Unit vectors are vectors with a magnitude of 1 that are used as a reference in expressing other vectors. In this context, unit vectors are typically denoted as i, j, and k, where i represents the x-axis unit vector, j represents the y-axis unit vector, and k represents the z-axis unit vector (if applicable). These unit vectors indicate the direction of the x, y, and z axes, respectively.
To express vectors A, B, and C in terms of unit vectors, you need to determine their x, y, and z components. This can be done using trigonometry and the given angles and magnitudes. Let's start with vector A.
For vector A:
- The angle of 37 degrees indicates the angle between the vector and the positive y-axis.
- To find the x and y components of vector A, you can use trigonometric functions.
- The x component of vector A is obtained by multiplying the magnitude (12 m) by the cosine of the angle: 12 * cos(37) = A_x.
- The y component of vector A is obtained by multiplying the magnitude (12 m) by the sine of the angle: 12 * sin(37) = A_y.
- So, vector A can be expressed as A = A_x * i + A_y * j.
Similarly, you can find the x and y components for vectors B and C using the given angles and magnitudes. Remember to consider the appropriate angle measurements based on the quadrant in which the vector is located.
Now let's move on to finding vectors R = A + B + C and S = C – A – B using unit vector notation.
To find vector R:
- Sum the x components of vectors A, B, and C to get the x component of R.
- Sum the y components of vectors A, B, and C to get the y component of R.
- Sum the z components of vectors A, B, and C (if applicable) to get the z component of R.
- Express vector R as R = R_x * i + R_y * j + R_z * k (if applicable).
To find vector S:
- Subtract the x component of vector A from the x component of vector C to get the x component of S.
- Subtract the y component of vector A from the y component of vector C to get the y component of S.
- Subtract the z component of vector A from the z component of vector C (if applicable) to get the z component of S.
- Express vector S as S = S_x * i + S_y * j + S_z * k (if applicable).
Finally, to determine the magnitude and direction of vectors R and S:
- The magnitude of a vector can be calculated using the formula: √(x^2 + y^2 + z^2) (if applicable).
- The direction of a vector is typically expressed as an angle measured counterclockwise from the positive x-axis (for 2D vectors) or as angles with respect to the positive x, y, and z axes (for 3D vectors).
By following these steps, you should be able to express the vectors A, B, and C in terms of unit vectors, find vectors R and S using unit vector notation, and determine their magnitudes and directions.