Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: A vector describes a displacement or force by its length and direction, recorded as the component pair ⟨a, b⟩. Its magnitude is the square root of a² + b², vectors add component by component, and the dot product u·v = u₁v₁ + u₂v₂ equals |u||v| cos θ, so it gives the angle between vectors and is zero for perpendicular vectors.
Standards: N-VM.A.1, N-VM.A.2, N-VM.A.3, N-VM.B.4, N-VM.B.5 · Level: Grade 12 Precalculus
What is a vector and how do you describe it?
A vector has both magnitude (length) and direction, unlike a scalar, which has only size. In component form a vector is v = ⟨a, b⟩, showing the horizontal and vertical change. From an initial point (x1, y1) to a terminal point (x2, y2), the components are ⟨x2 − x1, y2 − y1⟩.
- Magnitude: |v| = √(a2 + b2).
- Direction angle: tan θ = b/a, adjusted for the quadrant.
- From magnitude m and angle θ: ⟨m cos θ, m sin θ⟩.
How do you add, subtract and scale vectors?
Add and subtract components: ⟨a, b⟩ ± ⟨c, d⟩ = ⟨a ± c, b ± d⟩. Geometrically, place the vectors tail to tip, or use the parallelogram rule. A scalar multiple kv = ⟨ka, kb⟩ has length |k| |v| and reverses direction if k < 0. The unit vector in the direction of v is v/|v|.
A plane flies at ⟨240, 0⟩ mph (due east) in a 50 mph wind blowing toward the northwest, ⟨−35.4, 35.4⟩. The ground velocity is ⟨204.6, 35.4⟩, with speed √(204.62 + 35.42) ≈ 207.7 mph and direction ≈ 9.8° north of east.
What does the dot product tell you?
- Compute u · v = u1v1 + u2v2.
- Find |u| and |v|.
- Use cos θ = u · v ÷ (|u| |v|), then θ = arccos of that value.
For u = ⟨5, 1⟩ and v = ⟨2, 4⟩, u · v = 10 + 4 = 14, |u| = √26 and |v| = √20, so cos θ = 14/22.80 and θ ≈ 52.1°. A dot product of 0 means the vectors are perpendicular (orthogonal).
Treating the dot product as a vector. It multiplies matching components and adds them, so the result is a single number.
Sketch the vectors before computing. A quick picture tells you whether an angle should be acute or obtuse, which catches sign errors.
- Components from points: ⟨x₂ − x₁, y₂ − y₁⟩; magnitude √(a² + b²).
- From magnitude m and angle θ: ⟨m cos θ, m sin θ⟩.
- Add vectors by adding components; scalar multiplication scales every component.
- Unit vector: v/|v|, a vector of length 1 in the same direction.
- Dot product: u·v = u₁v₁ + u₂v₂ = |u||v| cos θ, and u·v = 0 means perpendicular.
Working with Vectors in the Plane: frequently asked questions
What is the difference between a vector and a scalar?
A scalar is a single number, such as a temperature or a speed of 60 mph. A vector has a size and a direction, such as a velocity of 60 mph toward the northeast, and is written with components.
How do you find the magnitude and direction of a vector?
Use the Pythagorean Theorem for the magnitude, √(a² + b²), and the inverse tangent of b/a for the direction angle. Check the quadrant, because a calculator’s arctan only returns angles between −90° and 90°.
How do you add two vectors?
Add the horizontal components and add the vertical components. Geometrically, place the tail of the second vector at the tip of the first, and the sum goes from the first tail to the second tip.
What does the dot product tell you?
It measures how much two vectors point the same way. Its value equals |u||v| cos θ, so it can be used to find the angle between vectors. A dot product of zero means the vectors are perpendicular.
How do you find a unit vector?
Divide the vector by its magnitude. For ⟨3, 4⟩ the magnitude is 5, so the unit vector is ⟨3/5, 4/5⟩. The result has length 1 and points in the original direction.
Mis à jour le