If u = ⟨3, −2⟩ and v = ⟨1, 5⟩, then u + v = ?
Given a = ⟨4, −1⟩ and b = ⟨−3, 2⟩, the vector 2a − b equals:
If u = ⟨2, −1⟩ and v = ⟨−5, 4⟩, what is the magnitude of the vector 2u + v ?
Vectors live in the Number & Quantity domain of ACT Math, and most students find them friendlier than they expect. A vector is just a quantity with both a magnitude (size) and a direction — think of an arrow on a grid. On the ACT, vectors are usually written in component form like ⟨a, b⟩, sometimes as ⟨a, b⟩ or â + bĵ notation.
Nearly every ACT vector question reduces to two ideas: do arithmetic component by component, and find length using the Pythagorean theorem. If you can handle ordered pairs and a square root, you can handle ACT vectors.
This guide covers component arithmetic, scalar multiplication, and magnitude, then gives you three practice questions — easy, medium, and hard.
To add or subtract vectors, combine their matching components: the x-parts together and the y-parts together. To multiply a vector by a number (a scalar), multiply each component by that number.
2u − v where u = ⟨3, 1⟩, v = ⟨−2, 4⟩.
Scale: 2u = ⟨6, 2⟩. Subtract v: ⟨6 − (−2), 2 − 4⟩ = ⟨8, −2⟩.
2u − v with u = ⟨3, 1⟩, v = ⟨−2, 4⟩.
Writing 6 − 2 = 4 for the x-part forgets that subtracting −2 adds 2. Sign care on negative components is essential.
When you multiply a vector by a scalar, the scalar hits both components. A frequent wrong answer scales the x-part but leaves the y-part untouched, or vice versa.
The magnitude of a vector is its length — the straight-line distance from its tail to its tip. Because the components form the legs of a right triangle, the magnitude is just the hypotenuse.
Example: the magnitude of ⟨5, 12⟩ is √(25 + 144) = √169 = 13.
| Task | Rule | The Step Students Miss |
|---|---|---|
| Add / Subtract | ⟨a, b⟩ ± ⟨c, d⟩ = ⟨a ± c, b ± d⟩ | Sign errors when a component is negative. |
| Scalar multiply | k⟨a, b⟩ = ⟨ka, kb⟩ | Scaling only one of the two components. |
| Magnitude | | ⟨a, b⟩ | = √(a² + b²) | Forgetting to find the resultant first when a sum or difference is involved. |
Usually no. Most ACT vector questions are given in component form, so you can add, subtract, scale, and find magnitude using only arithmetic and the Pythagorean theorem. A direction angle occasionally appears, in which case basic right-triangle trig (or a calculator's inverse tangent) handles it, but pure component work covers the large majority of questions.
It is the same calculation as the distance between two points and the same idea as the absolute value (modulus) of a complex number a + bi, which is √(a² + b²). Recognizing that these are all the Pythagorean theorem in disguise means one skill earns you points across several question types.
If u = ⟨3, −2⟩ and v = ⟨1, 5⟩, then u + v = ?
Given a = ⟨4, −1⟩ and b = ⟨−3, 2⟩, the vector 2a − b equals:
If u = ⟨2, −1⟩ and v = ⟨−5, 4⟩, what is the magnitude of the vector 2u + v ?
Correct: A · Easy
Add component by component: x-parts 3 + 1 = 4; y-parts −2 + 5 = 3. So u + v = ⟨4, 3⟩.
Correct: B · Medium
Scale first: 2a = ⟨8, −2⟩. Now subtract b = ⟨−3, 2⟩: x-part: 8 − (−3) = 11. y-part: −2 − 2 = −4. Result: ⟨11, −4⟩.
Correct: A · Hard
Find the resultant first. 2u = ⟨4, −2⟩. Add v = ⟨−5, 4⟩: x-part: 4 + (−5) = −1. y-part: −2 + 4 = 2. So 2u + v = ⟨−1, 2⟩. Magnitude: √((−1)² + 2²) = √(1 + 4) = √5.