Vectors and Parametric Equations Quiz

Grade 11 · Math · Quiz

Test your understanding of vector operations, dot products, and parametric representations of curves.

Questions

1. If u = ⟨1, -2, 3⟩ and v = ⟨4, 0, -1⟩, what is u · v?

   1. 5
   2. 1
   3. -2
   4. 7
   Show answer

   Answer: 1

   u · v = (1)(4) + (-2)(0) + (3)(-1) = 4 + 0 - 3 = 1.

2. What is the unit vector in the direction of v = ⟨0, 5⟩?

   1. ⟨0, 5⟩
   2. ⟨0, 1⟩
   3. ⟨1, 0⟩
   4. ⟨5, 0⟩
   Show answer

   Answer: ⟨0, 1⟩

   |v| = 5, so v̂ = ⟨0, 5⟩/5 = ⟨0, 1⟩.

3. The parametric curve x = 3cos(t), y = 3sin(t) represents:

   1. A parabola
   2. An ellipse
   3. A circle of radius 3
   4. A line
   Show answer

   Answer: A circle of radius 3

   x² + y² = 9cos²(t) + 9sin²(t) = 9, which is a circle of radius 3.

4. What angle do vectors u = ⟨1, 0⟩ and v = ⟨1, 1⟩/√2 make with each other?

   1. 30°
   2. 45°
   3. 60°
   4. 90°
   Show answer

   Answer: 45°

   cos(θ) = u·v / (|u||v|) = (1/√2)/1 = 1/√2, so θ = 45°.

5. Which pair of vectors is parallel?

   1. ⟨1, 2⟩ and ⟨2, -1⟩
   2. ⟨3, 6⟩ and ⟨1, 2⟩
   3. ⟨1, 0⟩ and ⟨0, 1⟩
   4. ⟨2, 3⟩ and ⟨3, 2⟩
   Show answer

   Answer: ⟨3, 6⟩ and ⟨1, 2⟩

   ⟨3, 6⟩ = 3⟨1, 2⟩, so they are scalar multiples and therefore parallel.

6. The Cartesian form of x = t², y = t³ can be written as:

   1. y = x^(3/2)
   2. y = x^(2/3)
   3. y² = x³
   4. x = y^(2/3)
   Show answer

   Answer: y² = x³

   From y = t³ and x = t², we have t = x^(1/2), so y = (x^(1/2))^3 = x^(3/2)... but eliminating properly: t = y^(1/3) and x = (y^(1/3))^2 = y^(2/3). Equivalently y² = x³.

7. The dot product of two vectors is always a vector.

   Show answer

   Answer: False

   The dot product of two vectors is a scalar (a single number), not a vector.

8. The magnitude of vector v = ⟨6, 8⟩ is ___.

   Show answer

   Answer: 10

   |v| = √(6² + 8²) = √(36 + 64) = √100 = 10.

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