Vectors: Quantities With Direction

Some quantities need more than a number to describe them. Velocity isn't just "50 mph" — it's "50 mph north." Force has a strength and a direction. These are vectors: quantities with both magnitude and direction, usually drawn as arrows. The trick to working with them is to break each vector into its horizontal and vertical components, which turns vector arithmetic into ordinary arithmetic plus a dash of the Pythagorean theorem.

This guide covers components, magnitude, and vector operations, with worked and practice problems matched to real ACT difficulty at Northside Tutoring.

Why Vectors Matter

Vectors appear on the ACT and are foundational in Pre-Calculus and physics (velocity, force, displacement). The component approach connects them to the coordinate plane and the Pythagorean theorem you already know. (Beyond the SSAT; rarely on the SAT.)

What a Vector Is

A vector has both magnitude and direction — pictured as an arrow whose length is the magnitude and whose way it points is the direction. A scalar, by contrast, is just a number with no direction (like temperature or mass). Velocity is a vector; speed is its scalar magnitude.

Components

Any vector can be described by its components — how far it goes horizontally and vertically — written like ⟨4, 3⟩. The components are all you need to compute with the vector.

Finding the Magnitude

The magnitude (length) of a vector is found with the Pythagorean theorem on its components:

For ⟨4, 3⟩: |v| = √(16 + 9) = √25 = 5. The components and the magnitude form a right triangle.

Adding & Subtracting

To add or subtract vectors, just combine their matching components:

Geometrically, adding vectors is placing them tip-to-tail; the components make it pure arithmetic.

Scalar Multiplication

Multiplying a vector by a number (scalar) scales each component — stretching or shrinking the arrow without changing its direction (unless the scalar is negative, which reverses it). 2⟨3, 4⟩ = ⟨6, 8⟩.

Where You'll See This — Test by Test

No reference sheet covers vectors. The ACT tests components, magnitude, and basic operations; they're core Pre-Calculus and physics. Rarely on the SAT, and beyond the SSAT.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

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