How can I visualise vectors in 3D?

Plot each position vector from the origin, shade the triangle or parallelogram the vectors span, and add the cross product as a normal. A 3D view that you can rotate shows at once whether a vector is perpendicular to the plane.

Can I use Desmos or GeoGebra for M2 vectors?

Yes, both can draw vectors in 3D, but you enter each object yourself. SketchMath is built on GeoGebra and generates the whole construction from the question text, which is quicker when you just want to see the figure for a past-paper question.

Is the distance from a point to a plane in the M2 syllabus?

Plane equations are not, but the distance can be asked as an application of projection: project a vector from the plane to the point onto the normal n = AB × AC.

How to Visualise 3D Vectors for DSE M2

Most marks lost in M2 vector questions come from not seeing the figure: which vectors lie in the plane, which one is the normal, and where the foot of the perpendicular falls. Fix the picture first. Draw the points, join the two vectors that span the plane, put the cross product on as the normal, and only then calculate.

Draw my vector question in 3D

The example above in 3D: plane ABC, its normal n = 6i + 3j + 2k, and the perpendicular from P to its foot F. Rotating the view shows PF is parallel to n. Try it yourself →

Why M2 Vector Questions Feel Hard

A vector question describes a three-dimensional figure in one paragraph of text, and you have to work on flat paper. Three things go wrong:

The wrong pair of vectors. For an area or a normal you need two vectors from the same point of the plane. Without a picture it is easy to mix AB with CA.
Direction. Whether a × b points up or down, and whether a projection is positive or negative, cannot be seen from the numbers.
No way to check. On paper a wrong normal looks as plausible as a right one. In a 3D view a wrong normal visibly leans.

A Four-Step Routine for Any 3D Vector Question

Plot the points. Write each as a position vector from O.
Name the plane. Pick one point of the plane as the base and form two vectors from it, such as AB and AC.
Get the normal. n = AB × AC. Check it with two dot products.
Project. A distance to the plane is the projection of a joining vector onto n; an angle comes from a dot product; an area is half the magnitude of the cross product.

Then open the same question as a 3D graph and rotate it until the plane is edge-on. If your normal really is perpendicular, it will stand straight up from the edge.

Typing the Question Instead of Building the Graph

General graphing tools are powerful, but for a vector question you have to enter every point, vector and plane yourself, and a slip of the mouse can undo the setup. SketchMath works the other way round: you type the question (or upload a photo), and it writes the GeoGebra construction for you, with the points labelled as in the question. You can still drag and rotate the result.

Worked Examples — How to Visualise 3D Vectors

Example 1: Distance from a point to a plane, with the picture

Find the distance from P(1, 1, 5) to the plane through A(1, 0, 0), B(0, 2, 0) and C(0, 0, 3).

AB = −i + 2j, AC = −i + 3k
Normal: n = AB × AC = 6i + 3j + 2k, |n| = √(36 + 9 + 4) = 7
AP = j + 5k
AP · n = (0)(6) + (1)(3) + (5)(2) = 13
Distance = |AP · n| / |n| = 13/7

Answer: 13/7 (≈ 1.86)

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Cross product in 3D)Draw the vectors a = i + 2j + 2k and b = 2i − j + 2k in 3D and show a × b.

(Triangle in 3D)A(2, 0, 0), B(0, 3, 0), C(0, 0, 6). Show triangle ABC in 3D and find its area.

(Projection in 3D)Show the projection of b = 2i + 2j + k onto a = i + j in 3D.

"Basically every 3D topic (Core and M2) is a real help, and for some of the more complicated 2D questions and the four centres of a triangle it is pretty good too. With Desmos I often move something by accident and have to start again to change it, which is inconvenient and wastes time, and it cannot give you the whole graph just from typing in the question the way SketchMath does. The M2 vectors part will definitely bring in more users, because every one of those questions is genuinely tricky."

— A DSE student who uses SketchMath (user feedback, translated from Cantonese)