Add, subtract and scale vectors
Grade 10 Math Worksheet - Challenge Me
Circle the best answer.
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 170 km/h and keeps heading east, since just the headwind part counts
- 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
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 = ⟨15, 9⟩ has length about 17.5
- 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 = ⟨9, 1⟩ has length 82, not 8
A sled track passes through P(1, 2) and Q(7, 5). Ari places flag R(13, k) on the same straight line. Which k works?
- k = 11
- k = 8
- k = 6
- k = 10
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 + BC = AD + DC = ⟨10, 7⟩, so the sides add the same
- No: AB = ⟨6, 2⟩ but DC = ⟨7, 2⟩, so opposite sides are not equal vectors
- 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
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 = -5; they point opposite ways, because the multiplier is -1/4
- k = 5; they point the same way, because 20 ÷ 4 = 5
- k = -80; they point opposite ways, because -12 ÷ 3 = -4
- k = -5; they point the same way, because a multiplier below 1 just shrinks
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?
- ⟨-1, 4⟩, which is v − ½u
- ⟨5, 4⟩, which is v + ½u
- ⟨4, 2⟩, which is ½(u + v)
- ⟨7, 2⟩, which is u + ½v
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 = ⟨3, 4⟩ and v = ⟨2, -3⟩
- u = ⟨2, -3⟩ and v = ⟨3, 4⟩
- u = ⟨3, 4⟩ and v = ⟨-2, 3⟩
- u = ⟨6, 8⟩ and v = ⟨4, -6⟩
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?
- 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
- Not constant: the hourly slides are ⟨4, 3⟩, ⟨4, 3⟩ and ⟨4, 4⟩, so hour 3 differs
- Constant: the run grows by exactly 4 km in each hour
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?
- Add the corners: P + Q + R = (12, 9), which is not G, so the claim fails
- ½ of side PQ is ⟨4.5, 1.5⟩, 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
- Find M = (6, 4.5); ½ of that is (3, 2.25), not G, so the claim fails
Read this first
Use this
|u| = 5, |v| = 10. u + v = ⟨-2, 11⟩, so |u + v|² = 4 + 121 = 125.
Luz tests u = ⟨4, 3⟩ and v = ⟨-6, 8⟩. Their lengths are 5 and 10, and |u + v| = √125. What does 5² + 10² = 125 show?
- u and v are perpendicular, so u + v closes a right triangle with legs 5 and 10
- Nothing: 125 is a coincidence that depends on these particular numbers
- u and v point the same way, so the lengths add and |u + v| = 15
- u and v point opposite ways, so |u + v| = 10 − 5 = 5, the lengths' difference