Vectors Quiz

Grade 12 · Math · Quiz

10 questions on vector operations, dot products, and geometric interpretations.

Questions

1. What is <3, 1> + <-2, 5>?

   1. <1, 6>
   2. <5, -4>
   3. <1, -4>
   4. <6, 5>
   Show answer

   Answer: <1, 6>

   Add components: <3+(-2), 1+5> = <1, 6>.

2. Find the magnitude of v = <-5, 12>.

   Show answer

   ||v|| = sqrt(25 + 144) = sqrt(169) = 13.

3. u = <1, 2>, v = <3, -1>. What is u · v?

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

   Answer: 1

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

4. Scalar multiplication changes the direction of a vector when the scalar is negative.

   Show answer

   Multiplying by a negative scalar reverses the direction of the vector.

5. Find the angle between u = <1, 0> and v = <0, 1>.

   1. 0°
   2. 45°
   3. 90°
   4. 180°
   Show answer

   Answer: 90°

   u·v = 0, so cos θ = 0, giving θ = 90°.

6. What is 3 times the vector <2, -4>?

   Show answer

   Multiply each component by 3: <6, -12>.

7. Which of the following is a unit vector?

   1. <1, 1>
   2. <1/√2, 1/√2>
   3. <2, 0>
   4. <1, 0, 1>
   Show answer

   Answer: <1/√2, 1/√2>

   Magnitude: sqrt(1/2 + 1/2) = 1. That is the only unit vector listed.

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

   Show answer

   The dot product produces a scalar, not a vector.

9. Find a vector parallel to <4, -2> with magnitude 1.

   Show answer

   ||<4,-2>|| = sqrt(16+4) = sqrt(20) = 2sqrt(5). Unit vector: <4/(2sqrt5), -2/(2sqrt5)> = <2/sqrt5, -1/sqrt5>.

10. If v = <a, b> and ||v|| = 1, what must be true?

    1. a = b
    2. a + b = 1
    3. a² + b² = 1
    4. a = 1 and b = 0
    Show answer

    Answer: a² + b² = 1

    ||v||² = a² + b² = 1 for a unit vector.

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