Vectors and Parametric Equations
Grade 11 · Math · Study Guide
An introduction to vector operations and representing curves through parametric equations in two and three dimensions.
A vector is a quantity with both magnitude and direction, represented as an ordered set of components such as v = ⟨a, b⟩ in 2D or v = ⟨a, b, c⟩ in 3D. Vectors support addition, scalar multiplication, and subtraction. The magnitude of vector v = ⟨a, b⟩ is |v| = √(a² + b²). The dot product of two vectors u = ⟨a, b⟩ and v = ⟨c, d⟩ is u · v = ac + bd, which equals |u||v|cos(θ), allowing calculation of the angle between vectors. Two vectors are perpendicular when their dot product is zero.
Parametric equations express the coordinates of a curve as functions of a parameter, typically t. For example, x = cos(t) and y = sin(t) trace a unit circle. Parametric form allows representation of curves that are not functions (fail the vertical line test) and provides directional information. To convert parametric equations to a Cartesian equation, eliminate the parameter t algebraically.
Vectors are fundamental in physics, computer graphics, and navigation. They are used to describe displacement, velocity, and forces. Understanding direction vectors and position vectors allows us to write parametric equations for lines: a line through point P_0 = (x_0, y_0) with direction vector d = ⟨a, b⟩ has parametric form x = x_0 + at, y = y_0 + bt.
Key terms
- Vector
- A mathematical object with both magnitude and direction, represented by components in coordinate form.
- Dot Product
- For vectors u and v, u · v = |u||v|cos(θ); equals the sum of products of corresponding components.
- Magnitude
- The length of a vector; for v = ⟨a, b⟩, |v| = √(a² + b²).
- Parametric Equations
- A set of equations that express coordinates as functions of a parameter t, used to describe curves.
- Unit Vector
- A vector with magnitude 1, found by dividing a vector by its magnitude; v̂ = v/|v|.
- Direction Vector
- A vector indicating the direction of a line; used in parametric equations for lines.
Flashcards
- Vector addition
- ⟨a, b⟩ + ⟨c, d⟩ = ⟨a+c, b+d⟩
- Scalar multiplication
- k⟨a, b⟩ = ⟨ka, kb⟩
- Dot product formula
- u · v = a₁a₂ + b₁b₂; also equals |u||v|cos(θ)
- Perpendicular vectors
- Two vectors are perpendicular (orthogonal) if and only if their dot product is 0
- Unit vector formula
- v̂ = v / |v| = ⟨a, b⟩ / √(a²+b²)
- Parametric line
- Through (x_0, y_0) with direction ⟨a, b⟩: x = x_0 + at, y = y_0 + bt
- Eliminating parameter
- Solve one equation for t and substitute into the other to get a Cartesian equation
- Circle parametrically
- x = r·cos(t), y = r·sin(t) traces a circle of radius r
Questions
What is the magnitude of the vector v = ⟨3, -4⟩?
- 1
- 5
- 7
- 25
Show answer
Answer: 5
|v| = √(3² + (-4)²) = √(9 + 16) = √25 = 5.
The dot product of u = ⟨2, 3⟩ and v = ⟨4, -1⟩ is ___.
Show answer
Answer: 5
u · v = (2)(4) + (3)(-1) = 8 - 3 = 5.
Vectors u = ⟨1, 2⟩ and v = ⟨-4, 2⟩ are perpendicular.
Show answer
Answer: True
u · v = (1)(-4) + (2)(2) = -4 + 4 = 0, so they are perpendicular.
The parametric equations x = 2 + 3t and y = 1 - t describe a line through point (2, 1) with direction vector:
- ⟨2, 1⟩
- ⟨3, -1⟩
- ⟨-3, 1⟩
- ⟨1, 3⟩
Show answer
Answer: ⟨3, -1⟩
The direction vector is the coefficient of t in the parametric equations: ⟨3, -1⟩.
Eliminating t from x = t + 1 and y = 2t - 3 gives the Cartesian equation y = 2x - ___.
Show answer
Answer: 5
From x = t + 1, t = x - 1. Substituting: y = 2(x-1) - 3 = 2x - 2 - 3 = 2x - 5.
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