ACT Math Vectors: 11 Practice Problems with Step-by-Step Explanations
Vectors on the ACT are usually written in one of two equivalent forms: component form, like ⟨5,−3⟩, or unit vector form, like 5i−3j, where i and j represent one unit in the x and y directions respectively. Both forms describe the exact same thing — a quantity with both a size (magnitude) and a direction — and most problems only require basic arithmetic once you're comfortable moving between the two notations.
Adding, subtracting, and scaling vectors all work component by component: to add two vectors, add their x-components together and their y-components together separately; to scale a vector by a number, multiply every component by that number. Real-world velocity and direction problems (a bus, car, or person moving at some angle) get translated into vector components using right-triangle trigonometry — an object moving at speed r in a direction θ from east has horizontal component r·cos(θ) and vertical component r·sin(θ). A vector's magnitude — its length, independent of direction — is found the same way as distance between two points: √(x²+y²), and importantly, a vector's magnitude never changes just because you rotate or redirect it, since magnitude only depends on speed or size, not direction.
Common traps include mixing up which direction is "positive" (east and north are typically positive, so west and south are negative), forgetting to distribute a scalar across every component (not just one), and on problems asking for a point along a vector at some fraction of its length, forgetting to add that fractional vector back to the starting point rather than treating it as the final answer on its own. Work through the 11 problems below, including one that references a figure, then click to reveal each step-by-step explanation.
On this page:
- Problem 1 — Vector subtraction
- Problem 2 — Vector addition
- Problem 3 — Scalar multiplication and addition
- Problem 4 — Solving for unknown components
- Problem 5 — Solving a vector equation
- Problem 6 — Magnitude of a vector
- Problem 7 — Velocity as a vector
- Problem 8 — Estimating a vector combination from a graph
- Problem 9 — Velocity from speed and direction
- Problem 10 — Finding a point along a vector
- Problem 11 — Displacement as a vector
Practice Problems
Given that u and v are vectors such that u = ⟨5,−3⟩ and v = ⟨2,4⟩, what is the component form of the vector u−v?
- A) ⟨7,−7⟩
- B) ⟨3,1⟩
- C) ⟨3,−7⟩
- D) ⟨−3,−7⟩
Subtract corresponding components: u−v = ⟨5−2, −3−4⟩ = ⟨3,−7⟩.
The vector 4i+5j is added to the vector −2i+6j. What is the sum of the two vectors?
- A) 2i−1j
- B) 2i+11j
- C) 4i+11j
- D) 6i+11j
Add corresponding components: (4+(−2))i + (5+6)j = 2i+11j.
Given that u and v are vectors such that u = ⟨7,1⟩ and v = ⟨2,8⟩, what is the component form of the vector 2u+3v?
- A) ⟨9,9⟩
- B) ⟨16,10⟩
- C) ⟨18,18⟩
- D) ⟨20,26⟩
Scale each vector first: 2u = ⟨14,2⟩, and 3v = ⟨6,24⟩.
Add them: ⟨14+6, 2+24⟩ = ⟨20,26⟩.
When the vector ai+4j is subtracted from the vector 3i+bj, the difference is 7i−3j. What are the values of a and b?
- A) a=−10 and b=3
- B) a=4 and b=−1
- C) a=10 and b=8
- D) a=−4 and b=1
Set up the equation: (3i+bj) − (ai+4j) = 7i−3j, which gives (3−a)i + (b−4)j = 7i−3j.
Match components: 3−a=7, so a=−4. And b−4=−3, so b=1.
Vector z can be expressed as ai+bj, vector w can be expressed as 6i+2j, and vector x can be expressed as 14i−j. If z+2w=x, a+b is equal to:
- A) 8
- B) 5
- C) 2
- D) −3
Substitute the given vectors: (ai+bj) + 2(6i+2j) = 14i−j, which gives (a+12)i + (b+4)j = 14i−j.
Match components: a+12=14, so a=2. And b+4=−1, so b=−5. Sum: a+b = 2+(−5) = −3.
Vince is drawing a vector on the standard (x,y) coordinate plane to show a bus driving 45° north of east at 36 miles per hour. If each coordinate unit represents 1 mile per hour, what is the magnitude of the vector?
- A) 18
- B) 18√2
- C) 36
- D) 36√2
A vector's magnitude is its length, which equals the original speed regardless of the direction it points. Rotating a vector to point at 45° instead of straight east doesn't change how long it is.
Since the bus is traveling at 36 mph, the vector's magnitude is 36.
The vector i represents 1 mile per hour east, and the vector j represents 1 mile per hour north. Sarah is jogging west at 9 miles per hour. Which of the following vectors represents Sarah's velocity, in miles per hour?
- A) −9i
- B) −9j
- C) 9i
- D) 9i
Since i represents east, west is the negative direction along i (there's no north/south component since Sarah is jogging purely west).
Her velocity vector is −9i.
The vectors a, b, and c are represented in the standard (x,y) coordinate plane below. In what general direction will the vector b−a+c point?
- A) Up and to the right.
- B) Up and to the left.
- C) Down and to the right.
- D) Down and to the left.
Vector a points directly left (negative x, roughly zero y). Vector b points mostly upward with a slight rightward lean. Vector c points up and to the right at a moderate angle.
Computing b−a+c: subtracting a (which points left) is equivalent to adding a vector pointing right, and both b and c already contribute upward and rightward components. The combined result points up and to the right.
The vector i represents 1 mile per hour east, and the vector j represents 1 mile per hour north. According to his car's GPS, at a particular instant, James is driving 30° north of east at 50 miles per hour. Which one of the following vectors represents James' velocity, in miles per hour, at that instant?
- A) −25i+25√3j
- B) 25i−25√3j
- C) 25i+25√3j
- D) 25√3i+25j
The east component (i) uses cosine: 50cos(30°) = 50(√3/2) = 25√3. The north component (j) uses sine: 50sin(30°) = 50(1/2) = 25.
James' velocity vector: 25√3i + 25j.
Vector DE starts at point D(−1,4) and ends at point E(5,14). A student is asked to draw point F on DE such that DE=3DF. What should the student draw point F?
- A) (1, 6⅔)
- B) (1, 7⅓)
- C) (3, 10⅔)
- D) (3, 11⅓)
Vector DE = E−D = (5−(−1), 14−4) = (6,10). Since DE=3DF, vector DF = (1/3)DE = (2, 10/3).
Point F is found by adding vector DF to point D: F = (−1+2, 4+10/3) = (1, 12/3+10/3) = (1, 22/3) = (1, 7⅓).
In the coordinate plane, i represents 1 mile east and the vector j represents 1 mile north. Tia drives 5 miles south and 5√3 miles west. Which of the following vectors represents Tia's final location relative to where she began driving?
- A) −5i+5√3j
- B) −5√3i−5j
- C) −10i
- D) 5√3i+5j
West is the negative direction along i, and south is the negative direction along j. The distance traveled west is 5√3 miles, and the distance traveled south is 5 miles.
Combined vector: −5√3i − 5j.
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