Vectors Worksheets

What it is

A vector is a quantity with both magnitude (size) and direction, written in component form as ⟨x, y⟩ (or ⟨x, y, z⟩ in three-space). The dot product of two vectors gives a number and reveals the angle between them; the cross product (three-space only) gives a new vector, perpendicular to both originals.

Why it matters

Vectors describe quantities that have both size and direction, force, velocity, displacement, and are the language used across physics, engineering, and computer graphics for anything that isn't just a single number.

How to do it

1. To find the vector from point A to point B, subtract: components of B minus components of A.
2. Magnitude: |v| = √(x² + y²) (add z² too in three-space).
3. Dot product a·b = a₁b₁ + a₂b₂ (+ a₃b₃): multiply matching components, add the results.
4. Cross product (three-space only) gives a new vector using the formula (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁).

Examples by level

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

• Treating a vector like a plain number and dropping its direction.
• Mixing up the dot product (gives a number) with the cross product (gives another vector).
• Forgetting the cross product only works in three dimensions.
• Sign errors when converting between polar (magnitude/angle) and Cartesian (component) form.

Tips

• A vector's direction matters as much as its size, always keep the sign of every component.
• Dot product answer is one number; cross product answer is a whole new vector.
• Two vectors are perpendicular exactly when their dot product is 0.

Key vocabulary

magnitude

the length of a vector, written |v|

dot product

multiply matching components and add — gives a single number, used to find angles and check perpendicularity

cross product

an operation on two three-space vectors that produces a new vector perpendicular to both

unit vector

a vector with magnitude 1, found by dividing every component by the magnitude

Frequently asked questions

What's the difference between the dot product and the cross product?

The dot product of two vectors is a single number (a scalar); the cross product (3-space only) is itself a new vector, perpendicular to both original vectors.

How do you tell if two vectors are perpendicular?

Their dot product is exactly 0. This works in both 2-space and 3-space.

What is a unit vector?

A vector with magnitude 1, pointing in the same direction as the original vector, found by dividing every component by the vector's magnitude.