Cal Poly SLO Calc. 3 Final Exam Review 2

Vectors

100

Determine 5*u-4*v for the vectors u=<3, -4, 1> and v=<-2, 5, -6>.

What is <23, -40, 29>?

200

Determine a vector v that is 2 units long and in the opposite direction of u=<4, -1, 4>.

What is v=<-8/sqrt(33), 2/sqrt(33), -8/sqrt(33)>?

300

Determine the angle between the vectors u=<-1, 0, -1> and v=<1, 1, 2>.

What is 5*pi/6?

400

Calculate proj_vu for the vectors u=<1, 1, -5> and v=<2, 1, -1>.

What is <8/3, 4/3, -4/3>?

500

Determine the cross product uxv for the vectors u=<1, 1, -1> and v=<2, 1, 1>.

What is <2, -3, -1>?

Points, Lines, and Planes

100

Calculate the acute angle between the planes x+y=1 and y+z=1.

What is pi/3?

200

Find the distances between each combination of the following points: P=(1, -1, 2), Q=(2, 1, 3), and R=(-1, 2, -1).

What is |PQ|=sqrt(6), |PR|=sqrt(22), |QR|=sqrt(26)?

300

Find the area of the triangle with the following vertices: A=(2, -1, 4), B=(1, 0, -1), and C=(1, 2, 3).

What is 3*sqrt(6)?

400

Find the equation of the plane that passes through the following points: P=(1, -1, 2), Q=(2, 1, 3), and R=(-1, 2, -1).

What is -9x+y+7z=4?

500

Given the point P=(2, 2, 0), determine the distance from P to the line x =-t, y=t, and z=t-1 and the distance from P to the plane 3x-2y-4z=4.

What is sqrt(26/3) and sqrt(4/29), respectively?

Vector Functions

100

Determine the equation for the acceleration of the vector function r(t)=<(4/9)*(1+t)^3/2, (4/9)*(1-t)^3/2, 3+e^-2t>.

What is a(t)=<(1/3)*(1+t)^-1/2, (1/3)*(1-t)^-1/2, 4e^-2t>?

200

Determine the equation for the tangent line to the curve r(t)=<ln(t), (t-1)/(t+2), t*ln(t)> at t=1.

What is x=t, y=t/3, z=t?

300

Determine the position function that has an acceleration function of a(t)=<(3/4)*(1+t)^-1/2, -e^-t, -1/(t + 1)²> with initial conditions v(0)=<3/2, 1, 1> and r(0)=<0, 0, 1>.

What is r(t)=<(1 + t)^3/2-1, 1-e^-t, 1+ln(t+1)>?

400

Determine the unit tangent vector function, the normal vector function, and the curvature function for the curve r(t)=<6*sin(2t), 6*cos(2t), 5t>.

What is T(t)=<(12/13)*cos(2t), (-12/13)*sin(2t), 5/13>, N(t)=<-sin(2t), -cos(2t), 0>, and κ(t)=24/169?

500

Determine the tangential and normal accelerations of the vector function r(t)=<e^t*sin(t), e^t*cos(t), sqrt(7)*e^t>.

What is a_T(t)=3e^t and a_N(t)=sqrt(2)*e^t?