MATH 223: Exam 1

Point-normal Form

A(x-x_o) + B(y-y_o) + C(z-z_o) = 0

Given two vectors in a plane, find the normal vector to the plane.

Take the cross product. n⃗ = v⃗₁ x v⃗₂

Given a point and a normal vector, find the plane.

Use point-normal form.

How do you determine if a line intersects a plane?

Plug the parametric equation into the plane equation. If there is a t that satisfies the equation, the line intersects.

What is true if a line does not intersect the plane?

The line may be parallel to the plane or lie in a different plane altogether.

What are the three major forms of plane equations?

Point-normal form, Standard form, and Slope plane form

What does the cross product give you?

A vector perpendicular (orthogonal) to both vectors.

How do you determine if the result of the cross product points into or out of the page?

Right-hand Rule. Point fingers along a and curl them towards b. The direction of your thumb is the direction of the cross product.

How can you determine if two vectors are paralell?

Two vectors are parallel if their cross product is zero, or if one is a scalar multiple of the other.

True or False: a⃗ × b⃗ = − (b⃗ × a⃗)

True.

Magnitude of the cross product

∣a⃗ × b⃗∣ = ∣a⃗∣ ∣b⃗∣ sin⁡(θ)

How do you find the angle θ between two vectors a⃗ and b⃗ ?

cos(θ) = (a⃗ ⋅ b⃗) / (|a⃗| |b⃗|)

How do you know if two vectors are perpendicular?

Their dot product is zero.

What’s the component of a⃗ in the direction of b⃗ ?

comp_b(a⃗ ) = (a⃗ ⋅ b⃗) / (|b⃗|)

What is a level set?

A set of points where a function f(x,y) is constant.

What do contour lines represent?

Curves along which the function has the same value. A top-down view of a surface.

How do you estimate a partial derivative from a table of function values?

Use a difference quotient.

What is the distance from a point (x_o, y_o, z_o) to a plane Ax + By + Cz = D ?

What is the distance between two points in 3D?