Two vectors in 3D: what their sum, dot and cross products really are

How to use this simulation

1. Two arrows start at the origin O. Vector a is the primary-colour arrow and vector b the secondary-colour one. At the start a = (3, 0, 0) and b = (1, 2, 0). The grey lines are the x, y and z axes, and the faint grid is the floor, where z = 0.
2. Drag anywhere on the picture to turn the scene. It is a flat screen, so depth is shown by the way the axes and arrows swing round as you rotate. While play is on, the view also spins slowly by itself. Press pause to hold it still.
3. Use the six sliders to set the x, y and z parts of a and of b. Every reading updates at once, and each arrow moves to its new tip.
4. Choose “Sum” and an orange arrow runs from O to the far corner of a shaded parallelogram. That arrow is a + b. The dashed sides show b laid at the tip of a, and a laid at the tip of b: both routes end at the same point.
5. Choose “Dot product”. The green arrow is the shadow of b on a, and the arc marks the angle θ between the vectors. The readings give a · b and θ.
6. Choose “Cross product”. The orange arrow is a × b. It stands at right angles to both a and b, and the shaded parallelogram has the area the readout gives. If the arrow gets too long to fit, it is drawn shorter and the picture says so.

What is a vector?

Some quantities need only a number: your age, the price of a pen. Others need a direction as well. “Walk 5 metres” is incomplete; “walk 5 metres north” is a complete instruction. A vector carries both: how much, and which way. We draw it as an arrow.

In 3D we write a vector by its three components, how far it goes along x, along y and along z. So (3, 0, 0) is an arrow of length 3 pointing straight along the x axis. The length, or magnitude, comes from Pythagoras applied twice: |v| = √(x² + y² + z²).

Adding vectors: the parallelogram

To add two vectors, add the matching components. With a = (3, 0, 0) and b = (1, 2, 0) the sum is (4, 2, 0). No arrows needed, just three small additions.

The picture shows why that works. Walk along a, then along b, and you finish where the diagonal of the parallelogram ends. Walk b first, then a, and you land in the same place. That is why a + b = b + a.

• Subtracting is the same with minus signs: a − b is a plus the reverse of b.
• Scaling a vector by 2 doubles each component and doubles the length, with the direction unchanged.
• The length of the sum is not the sum of the lengths. Here |a| = 3 and |b| = 2.24, but |a + b| is shorter than 3 + 2.24, because the arrows do not point the same way.

How a flat screen shows 3D

There is no 3D library here, only a flat canvas and a bit of trigonometry. The scene is turned about the vertical z axis by one angle (the yaw) and tipped towards you by another (the pitch). Each point (x, y, z) is then squashed onto the screen, keeping its sideways position and its height.

Because nothing is drawn nearer or farther, parallel lines stay parallel (an orthographic view, like an engineer’s drawing). Rotation is what tells your eye which way is which. Drag a few times, and the flat arrows start to look solid.

The dot product: one number

Multiply matching components and add the three results: a · b = ax·bx + ay·by + az·bz. For the starting pair that is 3. Notice that the answer is a plain number, not a vector, which is why it is also called the scalar product.

There is a second formula for the very same number: a · b = |a| |b| cos θ, where θ is the angle between the arrows. Put the two together and you can find any angle in 3D without drawing it: cos θ = (a · b) / (|a| |b|).

For the starting pair this gives θ = 63.43°, which is what the Dot product mode reports.

• Positive dot product: the angle is less than 90°, the arrows lean the same way.
• Zero dot product: exactly 90°. The vectors are perpendicular (orthogonal).
• Negative dot product: the angle is more than 90°, the arrows lean opposite ways.

The dot product as a shadow

Shine a light straight down onto a, and b casts a shadow along a. The signed length of that shadow is (a · b) / |a|. The dot product is then |a| times the shadow, which is the same as |b| times the shadow of a on b.

In the simulation that shadow is the green arrow, and the dashed line drops from the tip of b to the end of the shadow at a right angle. Drag b round: the shadow flips to the other side of O when the angle passes 90°.

The cross product: a new arrow

The cross product a × b is a vector, not a number. Each component is built from the other two coordinates, as the formula below shows.

For the starting pair it is (0, 0, 6): straight up along z. That is no accident, because a and b both lie flat in the floor, and the only direction at right angles to the floor is up or down.

• a × b is perpendicular to both a and b. Check by taking its dot product with either: it is 0.
• Its direction follows the right-hand rule: curl the fingers of your right hand from a towards b and your thumb points along a × b.
• Its length |a × b| is the area of the parallelogram. For the starting pair the area is 6 square units, the shaded shape in Cross product mode.
• If a and b are parallel the parallelogram is flat, the area is 0, and the cross product is the zero vector.

Order matters for the cross product

Swap the vectors and the arrow flips: b × a = −(a × b). The dot product has no such twist, a · b = b · a. The unit vectors i, j, k along the three axes show the pattern in the cleanest form.

Dot or cross: which one when?

Both combine two vectors, but they answer different questions.

Try it in the simulation

Six quick experiments, a minute each.

• In Sum mode set b to (0, 3, 0). The shaded shape becomes a rectangle. Why are the sides at right angles?
• In Dot product mode make b equal to a. The angle is 0° and a · b = |a|².
• Set a to (2, 0, 0) and drag the b x-slider from 4 down to −4. Watch the dot product pass through zero and change sign.
• In Cross product mode make b a multiple of a, like (6, 0, 0). The area is zero and the arrow vanishes.
• In Cross product mode set a = (3, 0, 0) and b = (0, 2, 0). The arrow points up with length 6. Now swap the two values. Which way does the arrow go?
• Give both vectors a non-zero z. Rotate the view until a, b and a × b look like three corners of a box.

Solved problems

Six common questions. Every number below is computed with the same functions as the simulation.

Mistakes to avoid

These cost marks again and again.

• Calling a · b a vector. It is a number. If you got three components, you computed something else.
• Using the cross product in two dimensions. It lives in 3D (the 2D version is a single number, the z component).
• Forgetting the order in a cross product, and so getting the sign of the answer wrong.
• Mixing up sin θ and cos θ: the dot product has cos θ and the cross product has sin θ.
• Adding lengths instead of components when finding |a + b|.

Where vectors are used

Games and film use the dot product to decide whether a surface faces the light, and the cross product to find which way a surface points. Physics uses the cross product for torque and for the force on a charge moving in a magnetic field. Engineers use the dot product for work and power.

Behind a phone map, a robot arm or a drone’s flight controller is a pile of exactly these two operations, run millions of times a second.

Exam corner

For HSC higher maths and admission tests, remember these.

• For perpendicular, set a · b = 0. For parallel, set a × b = 0 (or one is a multiple of the other).
• cos θ = (a · b) / (|a| |b|). Write the formula first, then substitute.
• The area of a parallelogram is |a × b|; the area of a triangle is half that.
• Check a cross product by dotting it with a and b. Both should give 0.

Frequently asked questions