Vectors in the Plane
What is the formula for the magnitude of a vector v = ⟨a, b⟩?
||v|| = √(a² + b²)
What is the formula for the dot product of u = ⟨a, b⟩ and v = ⟨c, d⟩?
u · v = ac + bd
In parametric equations, what is the variable t called?
The parameter
In polar coordinates, what do r and θ represent?
r = distance from origin; θ = angle from positive x-axis
State the most fundamental Pythagorean identity.
sin²θ + cos²θ = 1
Find the magnitude of v = ⟨3, 4⟩.
√(9 + 16) = √25 = 5
Find u · v if u = ⟨1, 2⟩ and v = ⟨3, 4⟩.
(1)(3) + (2)(4) = 11
Eliminate the parameter: x = t, y = t + 3.
y = x + 3
Convert the polar point (2, 0) to rectangular coordinates.
(2, 0)
If sin θ = 3/5, find cos²θ.
16/25
Given v = ⟨1, 3⟩ and w = ⟨2, −1⟩, find v + w.
⟨3, 2⟩
Two vectors are orthogonal when their dot product equals what value?
0
Eliminate the parameter: x = t + 2, y = t − 1.
y = x − 3
Convert the polar point (4, π/2) to rectangular coordinates.
(0, 4)
State the identity involving tan θ and sec θ.
1 + tan²θ = sec²θ
Find the unit vector in the direction of v = ⟨−3, 4⟩.
⟨−3/5, 4/5⟩
Are u = ⟨2, −1⟩ and v = ⟨1, 2⟩ orthogonal? Show your work.
u · v = 2 − 2 = 0, so yes
What curve do x = cos t, y = sin t trace out as t goes from 0 to 2π?
A circle of radius 1: x² + y² = 1
Convert the rectangular point (3, 3) to polar form.
(3√2, π/4)
If tan θ = 2, find sec²θ.
5
Find the unit vector in the direction of v = ⟨−3, 4⟩.
⟨−3/5, 4/5⟩
Find the projection of u = ⟨3, 4⟩ onto v = ⟨1, 0⟩.
u·v = 3, ||v||² = 1, so proj = ⟨3, 0⟩
Eliminate the parameter from x = 2t, y = t². Write y in terms of x.
y = x²/4
Convert the polar equation r = 5 to rectangular form. What shape is it?
x² + y² = 25, a circle of radius 5
Simplify: (1 − sin²θ) / cos²θ
1