3D vector visualizer
The dot product is a number, the cross product is a new vector — and it's easy to lose track of which is which. So build the two vectors and watch. Change any component and the dot product, cross product and angle between them update instantly, from the exact formulas you'll write in the exam.
The rules you're seeing
Dot product tells you about angle
Since a·b = |a||b|cosθ, the sign of the dot product alone tells you if the angle is acute (positive), obtuse (negative), or exactly 90° (zero) — no arccos needed just to check "are these roughly aligned or opposed?"
Cross product makes a NEW perpendicular vector
Unlike the dot product (a scalar), a×b is a vector, always perpendicular to both a and b. Its direction follows the right-hand rule: curl the fingers from a to b, thumb points along a×b.
Cross product magnitude = parallelogram area
|a×b| equals the area of the parallelogram that a and b span — this is why the cross product shows up in torque, angular momentum, and area/volume calculations throughout physics and maths.
When vectors are parallel or perpendicular
a×b = 0 exactly when a and b are parallel (θ=0° or 180°); a·b = 0 exactly when they're perpendicular (θ=90°). Try setting b to a scalar multiple of a, or making them perpendicular, and watch each product vanish in turn.
Part of the Vector Algebra chapter — read the notes, grab the formula sheet and take the quiz. One of Priodemy for School, free with every EduSuite school.
Two ways to multiply vectors, and why both exist
The dot product measures alignment
The dot product takes two vectors and returns a single number: a · b = |a||b| cos θ, or in components a₁b₁ + a₂b₂ + a₃b₃. Because the cosine is largest when the vectors point the same way, the dot product is a measure of how much of one vector lies along the other.
That makes the sign informative on its own. Positive means the angle is acute, negative means obtuse, and zero means the vectors are perpendicular — the cleanest test for a right angle in three dimensions, needing no diagram. Rotate the vectors here and watch the value pass through zero exactly as they square up.
The dot product is why work is defined as F · d: only the component of force along the displacement does any work, and a force at right angles to the motion does none. That is the same fact seen earlier with magnetic forces.
The cross product makes a new vector
The cross product returns a vector rather than a number. Its magnitude is |a × b| = |a||b| sin θ, and its direction is perpendicular to both inputs, with the sense given by the right-hand rule. Where the dot product peaks for parallel vectors, the cross product vanishes for them, since sin 0° = 0. So a × b = 0 is the test for parallel, just as a · b = 0 is the test for perpendicular.
The magnitude has a direct geometric reading: it equals the area of the parallelogram spanned by the two vectors, so half of it is the area of the triangle they form. That is why cross products appear in questions asking for areas in three dimensions, where the usual half-base-times-height is awkward to apply.
Order matters. Swapping the operands reverses the result: a × b = −(b × a). The dot product has no such issue, being commutative. This asymmetry catches people out, and it is worth checking the order before computing.
Mistakes that cost marks
Returning the wrong kind of object. A dot product is a scalar and a cross product is a vector. Writing a · b as a vector, or a × b as a plain number, loses marks even when the arithmetic is right.
Reversing the cross product. Because a × b = −(b × a), a normal vector computed in the wrong order points the opposite way. In plane equations this flips the sign of every term.
Forgetting to normalise. Questions asking for a unit vector perpendicular to two others need the cross product divided by its own magnitude. Stopping at the cross product gives the right direction but the wrong length.