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.

Vectors · Grades 12–12

Vectors Worksheets

Grade 12 vector worksheets: representing vectors (converting between magnitude/direction and Cartesian components), operations (addition, subtraction, scalar multiplication, dot product, cross product), and their named applications, angle between vectors, perpendicularity, projection, work, and the area of a parallelogram.

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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

In a physics simulation, a single push has both a strength and a direction, split those apart and the physics falls apart too, which is exactly why a vector keeps them bundled as one object instead of two separate numbers.

How to do it

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

Examples by level

BeginnerFind $\vec{AB}$ given A(-6, 6) and B(3, 0) =Answer $\langle 9, -6 \rangle$
IntermediateFind $\vec{AB}$ given A(-7, 2) and B(-3, -3) =Answer $\langle 4, -5 \rangle$
Advanced $\vec{a} = \langle 2, 4 \rangle$ , $\vec{b} = \langle 0, -7 \rangle$ . Find $\vec{a} - \vec{b}$ =Answer $\langle 2, 11 \rangle$

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.

For parents

A vector is really two pieces of information glued together, how far and which way, so any answer should always carry both, a number, a set of components, or an angle.

For teachers

Ground the dot product in its two equivalent formulas (a·b = a₁b₁+a₂b₂ and a·b = |a||b|cosθ) early, most of this unit's applications (angle between vectors, perpendicularity, projection) are just that one relationship rearranged.

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.

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