Vectors
1. Find the angle between the planes with the given equations.
2x - y + z = 5 and x + y - z = 1
2. Find the values of r' (t) and r'' (t) for the given values of t.
r (t) = i cos t + j sin t; t = pi/4
3. The acceleration vector a (t), the initial position r = r (0), and the initial velocity v = v (0) of a particle moving in
xyz-space are given. Find its position vector r (t) at time t.
a(t) = 6ti - 5j + 12t²k; r = 3i + 4j; v = 4j - 5k
4. Find the curvature of the given plane curve at the indicated point.
10. x = t - 1, y = t² + 3t + 2, where t = 2
5. Find the unit tangent and normal vectors at the indicated point.
18. x = t³, y = t² at (-1, 1)
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Problem: Find the angle between the planes with the given equations:
2x - y + z = 5 and x + y - z = 1
Solution:
Angle between two planes (h_1,k_1,l_1,d_1) and (h_2,k_2,l_l,d_2) is given as
Cos∅=(h_1 h_2+k_1 k_2+l_1 l_2)/(√(h_1^2+k_1^2+l_1^2 )×√(h_2^2+k_2^2+l_2^2 ))
=((2)×(1)+(-1)×(1)+(1)×(-1))/(√((2)^2+(-1)^2+(1)^2 )×√((1)^2+(1)^2+(-1)^2 ))
=0
Or,Cos∅=0
Or,∅=〖Cos 〗^(-1) 0= 90°=π/2
Therefore the angle between the two given planes is π/2
Problem: Find the values of r' (t) and r'' (t) for the given values of t:
r (t) = i cos t + j sin t; t = π/4
Solution:
r (t)= i cost+ j sint
r^' (t)=d{r(t)}/dt=∂(Cos t)/∂t i+∂(Sin t)/∂t j=-Sin ...
Solution Summary
This solution is comprised of detailed step-by-step calculations and analysis of the given problems related to Vectors and provides students with a clear perspective of the underlying concepts.