Lecture 10-11

Question 1

Vector equation for a plane in r^n going through point p with direction d

Answer 1

[x1, x2, . . . , xn] = x = OP + td where OP is the vector from the origin to point p and t is a real number

Question 2

vector equation of a plane in r^n with point p

Answer 2

OP + su + tv

OP is the vector of point p, v & u are non-zero non-parallel vectors, and s & t are real numbers

Question 3

when is vector n a normal vector for a plane

Answer 3

if n is orthogonal to any vector PQ for any point p,q on the plane

Question 4

scaler equation for a plane with normal n = [a,b,c], and n * OP = d for any point p on the plane

Answer 4

scaler equation of a plane: ax1 +bx2 + cx3 = d

Question 5

n * OP = d when what conditions are true

Answer 5

P is a point on the hyperplane, n a normal vector of the hyperplane & n * OP = d

Question 6

proj_v(u) =? (projection of u onto v)

Answer 6

proj_v(u) = v(u * v)/(||v||^2)

Question 7

perp_v(u) = ? (perpendicular of u onto v)

Answer 7

perp_v(u) = u - proj_v(u)

Question 8

proj_v(u) * perp_v(u) = ? (dot product)

Answer 8

= 0

Question 9

Equation for the area of a parallelepiped

Answer 9

|a * (b x c)|