Add, subtract and scale vectors

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   Ground velocity is their sum. Noah's plane has airspeed vector ⟨240, 0⟩ km/h. The wind adds ⟨-70, 168⟩ km/h (182 km/h in all). Which statement about the plane's motion over the ground is correct?

   • It moves at 422 km/h, since the plane's 240 and the wind's 182 add
   • It moves at about 353 km/h, and it heads about 28° south of east
   • About 239 km/h, nearly its airspeed, but heading about 45° north of east
   • It moves at 170 km/h and keeps heading east, since just the headwind part counts
2. 2.

   A student claims |u − v| = |u| − |v|. With u = ⟨12, 5⟩ and v = ⟨3, 4⟩ that gives 13 − 5 = 8. Kira checks it. What does Kira find?

   • The claim fails: u − v = ⟨9, 1⟩ has length 82, not 8
   • The claim fails: u − v = ⟨9, 1⟩ has length √82, about 9.1, not 8
   • The claim holds: 13 − 5 = 8, and both lengths are exact whole numbers
   • The claim fails: u − v = ⟨15, 9⟩ has length about 17.5
3. 3.

   Tala's quilt has corners A(0, 0), B(6, 2), C(10, 7) and D(3, 5) in order. Tala asks if ABCD is a parallelogram. Which reasoning is correct?

   • Yes: AB and DC both rise 2 units, so those two sides must be parallel
   • No: the diagonals AC and BD have different lengths, and a parallelogram's must match
   • No: AB = ⟨6, 2⟩ but DC = ⟨7, 2⟩, so opposite sides are not equal vectors
   • Yes: AB + BC = AD + DC = ⟨10, 7⟩, so the sides add the same
4. 4.

   Pia's two rocket paths ⟨3, k⟩ and ⟨-12, 20⟩ lie along one line. Which k works, and do the paths point the same way?

   • k = -80; they point opposite ways, because -12 ÷ 3 = -4
   • k = 5; they point the same way, because 20 ÷ 4 = 5
   • k = -5; they point opposite ways, because the multiplier is -1/4
   • k = -5; they point the same way, because a multiplier below 1 just shrinks
5. 5.

   Zee's quilt is parallelogram ABCD with A at the origin, AB = u = ⟨6, 0⟩ and AD = v = ⟨2, 4⟩. M is the midpoint of side DC. Which vector is AM?

   • ⟨4, 2⟩, which is ½(u + v)
   • ⟨5, 4⟩, which is v + ½u
   • ⟨7, 2⟩, which is u + ½v
   • ⟨-1, 4⟩, which is v − ½u
6. 6.

   Two vectors satisfy u + v = ⟨5, 1⟩ and u − v = ⟨1, 7⟩. Jun's car and trailer move with u + v = ⟨5, 1⟩ and u − v = ⟨1, 7⟩. Which pair gives u and v?

   • u = ⟨2, -3⟩ and v = ⟨3, 4⟩
   • u = ⟨3, 4⟩ and v = ⟨2, -3⟩
   • u = ⟨3, 4⟩ and v = ⟨-2, 3⟩
   • u = ⟨6, 8⟩ and v = ⟨4, -6⟩
7. 7.

   Milo's drone log shows positions from the start of (4, 3), (8, 6) and (12, 10) km after 1, 2 and 3 hours. Milo claims constant velocity. What does the log show?

   • Constant: the run grows by exactly 4 km in each hour
   • Not constant: the hourly slides are ⟨4, 3⟩, ⟨4, 3⟩ and ⟨4, 4⟩, so hour 3 differs
   • Not constant: the distances from the start are 5, 10 and about 15.6 km, so they grow
   • Constant: the first two hours match, so the third reading must be a small error
8. 8.

   Esme's rug vector is u = ⟨-6, 8⟩. A vector ku is 20 long and points down and to the right. Which k works?

   • k = -2
   • k = 2 or k = -2, either one
   • k = 2
   • k = 20
9. 9.

   Kira's triangle has corners P(0, 0), Q(9, 3) and R(3, 6). Kira says G(4, 3) is two-thirds of the way along the median from P. Which check settles it?

   • ½ of side PQ is ⟨4.5, 1.5⟩, which is not G, so the claim fails
   • Find M = (6, 4.5); ½ of that is (3, 2.25), not G, so the claim fails
   • Add the corners: P + Q + R = (12, 9), which is not G, so the claim fails
   • Find M = ½(Q + R) = (6, 4.5); then ⅔ of ⟨6, 4.5⟩ is ⟨4, 3⟩, so the claim holds
10. 10.

    Bo's robot slides u = ⟨3, 4⟩, then turns a quarter turn left and slides the same distance. Where does it finish, as one slide from the start?

    • ⟨6, 8⟩
    • ⟨7, 1⟩
    • ⟨-1, 7⟩
    • ⟨7, 7⟩