The problem concern the position vectors r1=2.43mi-4.74mj and r2=-3.02mi+0.85mj.
Find the angle between the two vectors. Hint: Find the angle each makes with the +x-axis.
do the dot product, and the magnitude of the cross product.
cross product= r1XR2=magR1*magR2*sinTheta
dot product= magR1*magR2*cosTheta
cross product/dot product = tan theta
solve for theta.
How do you get cross product?
r1XR1=(2.3i-4.74j)X(3.02i+.85j)
= 2.31iX(3.02i+.85j) + (-4.74j)X(3.02i+.85j)
well iXi=0, iXj= k, jXi=-k, jXj=0
or = 2.31*.85k+4.74*3.02k=CCCCc
How do you get the dot product?
r1.r2= 2.3*3.02-4.74*.85=DDDDD
so tan theta = CCCCC/DDDDDD
I see I missed a sign on R2, correct that please.
To find the angle between the two vectors, first find the angle each vector makes with the +x-axis. Then, subtract one angle from the other to get the angle between them.
Step 1: Calculate the angle each vector makes with the +x-axis.
To find the angle that a vector makes with the +x-axis, you can use the following formula:
θ = arctan(y/x)
where θ is the angle, y is the y-component of the vector, and x is the x-component of the vector.
For r1 = 2.43mi - 4.74mj:
x1 = 2.43mi
y1 = -4.74mj
θ1 = arctan(-4.74/-2.43) = arctan(1.951) = 63.89 degrees
For r2 = -3.02mi + 0.85mj:
x2 = -3.02mi
y2 = 0.85mj
θ2 = arctan(0.85/-3.02) = arctan(-0.281) = -15.73 degrees
Note that we calculate the angle using the arctan function and not the arcsin function because we want the angle with respect to the +x-axis.
Step 2: Find the angle between the two vectors.
To find the angle between the two vectors, subtract one angle from the other:
θ = θ2 - θ1
θ = (-15.73) - 63.89 = -79.62 degrees
The angle between the two vectors is -79.62 degrees. Note that the negative sign indicates that the second vector is rotating clockwise relative to the first vector when observed from above the +x-axis.
Therefore, the angle between the two vectors is approximately -79.62 degrees.