Chapter 3.3 Flashcards
Def: Orthogonality (3 def)
Two vectors u and v in R^(n) are orthogonal (or perpendicular) if u·v=0. We will also agree that the zerovector in R^(n) is orthogonal to all vectors in R^(n). A nonempty set of unitvectors is called an orthonormal set.
How do you find the normal to a line in R² or a plane in R³?
1) If a and b are constants that are not both zero, then an equation:
ax + by + c = 0
Representes a line in R² with the normal n=(a,b)
2) If a, b, and c are constants that are not all zero, then an equation
ax + by + cz + d = 0
Represents a plane in R³ with normal n=(a,b,c)
How to decompose a vector v int oa sum of a vector a and a vector perpendicular to a
1) Drop a perpendicular from the top of u to the line through a
2) Construct the vector w1 from the initial point of u and a, codirectional with a to the foot of the perpendicular.
3) Construct the vector w2 = u - w1
If u and a are vectors in R^(n) and a != 0 then u can be expressed as the sum of the two following vectors:
w1 which is a scalar multiple of a
w2 which is orthogonal to a
What is the projection of u along a
proj(a)u=(u·a/||a||²)*a
What is the vector component of u orthogonal to a?
u - proj(a)u
What is the norm of the projection of u along a?
||proj(a)u|| = |u·a|/||a||
What is the norm of the projection of u along a if the angle between u and a are v
||proj(a)u|| = ||u||*|cos(v)|
Theorem of Pythagoras in R^(n)
||u + v||² = ||u||² + ||v||²
How do you find the distance between the point P(x0, y0) and the line
ax+by+c=0
ax0 + by0 + c | / sqrt(a² + b²)
How do you find the distance between point P(x0, y0, z0) and the plane ax + by + cz + d = 0
ax0 + by0 + c*z0 + d | / sqrt(a² + b² + c²)
How do you find an arbitrary point in the equation of a plane ax + by + cz + d = 0?
You set two variables to arbitrary values and calculate the value of the last variable.
