Test 1

Question 1

Finding an exact angle between two vectors

Answer 1

cos x = the dot product / the magnitudes multiplied

Question 2

Finding the area of a triangle made by three vectors

Answer 2

Combine one vector with each of the other two (i.e. PQ and PR) and find their cross product, then square each component and add them together under a radical and divide by 2

Question 3

Finding the equation of a sphere with a certain radius

Answer 3

Like a circle just with the z component

Question 4

Finding a unit vector in the direction a + b

Answer 4

Add the vectors together and divide by the magnitude of the two added vectors

Question 5

Finding projection of a onto b and its magnitude

Answer 5

a dot b / b dot b all multiplied by b; magnitude if found how it normally is after the projection is found

Question 6

Explaining the direction and magnitude of the cross product of any two 3d vectors

Answer 6

The direction is orthagonal to the vectors and the length is the area of the parallelogram created by the two vectors

Question 7

Finding the parallelepiped created by three vectors

Answer 7

Do it like a normal cross product but make the top vector the third vector instead of i + j + k

Question 8

Finding parametric equations of lines through two points

Answer 8

vector 2 + (vector 2 - vector 1)t then split it up into x=, y=, and z=

Question 9

Finding the equation of a plane containing three points

Answer 9

Combine one point with the two others to make two vectors then cross them; use the equation (x-x1) + (y-y1) + (z-z1) = 0 and multiply that with the vector found by crossing the vectors

Question 10

Finding an expression for the angle between two planes

Answer 10

cos^-1 (absolute value of the dot product of the two / both of the magnitudes multiplied together)

Question 11

Writing the parametric equation for the line of intersection of two planes

Answer 11

Find a point on the line using a system of equations then cross the coefficients of the two planes; write it like (x from point on plane) + (x from crossed vector)t for x, y, and z then transfer to fit with x=, y=, and z=

Question 12

Find the distance between R and a plane

Answer 12

Make a vector with the given point and a point on the plane then find the absolute value of the dot product of that point and the point made by the coefficients of the equation of the plane and divide that by the magnitude of the vector made by the coefficients of the equation of the plane

Question 13

Finding the distance between a point and a line

Answer 13

Use the vector attached to the t-values of the given line and cross that with the combination of the given point and a point on the given line; use that new vector and cross it with the combination vector; use that magnitude and divide it by the magnitude of the vector

Question 14

The shape created by the traces of two hyperbolas and an ellipse

Answer 14

elliptic hyperboloid

Question 15

The shape created by two parabolas and a hyperbolla

Answer 15

hyperbolic paraboloid

Question 16

The shape created by three ellipses

Answer 16

ellipsoid

Question 17

The shape made if one variable is infinite

Answer 17

cylinder

Question 18

Finding velocity, speed, and acceleration

Answer 18

Velocity is just the derivative of the vector and speed is its magnitude; acceleration is the derivative of velocity

Question 19

T, N, B and curvature

Answer 19

T= velocity / magnitude of velocity
N= T' / magnitude of T'
B= T x N
Curvature= magnitude of r' x r'' / magnitude of r' cubed

Question 20

Finding arclength

Answer 20

Derive the vector equation and then square each part under a radical and integrate that