Shunya / Vectors & 3D Geometry Lab

Shunya · Maths lab

Vectors & 3D geometry lab

See vectors instead of just calculating with them. Drag two vectors to add, scale and project them, turn a 3D scene to watch the dot product, the cross product and the volume of a box, then explore lines and planes in space: when lines meet, when they are skew, how far a point is from a plane and the angle between a line and a plane.

Lessons

Pick your class, follow the steps in the lab, and check what you see against the working under the graph.

Class 11 — Coordinates in three dimensions, distance and section formula

Concept: a point in space has three coordinates (x, y, z). The distance between A and B is √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²), the midpoint averages the coordinates, and the section formula splits a segment in a ratio m : n. The three coordinate planes cut space into eight octants.

Open Vectors in space and choose Distance & section formula.
Change A and B and read the distance: try A = (0, 0, 0) and B = (2, 3, 6) to get 7.
Set the ratio 1 : 1 and check that P is the midpoint M.
Set m : n = 2 : 1, then tick divide externally and see P move outside the segment.
Move a point through a coordinate plane and read which octant it is in.

Takeaway: the same distance and section formulas as in the plane, with one more coordinate.

Class 12 — Vectors: magnitude, direction and addition

Concept: a vector has magnitude and direction. The magnitude of xi + yj + zk is √(x² + y² + z²), the unit vector is a ÷ |a|, and the direction cosines l, m, n (the cosines of the angles with the axes) satisfy l² + m² + n² = 1. Vectors add component by component, by the triangle law or the parallelogram law.

Open Vectors in the plane and drag a and b: the sum is the diagonal of the parallelogram.
Tick also show a − b and press opposite vectors.
Choose Scalar & unit vector and change k: |k·a| = |k||a|.
Open Vectors in space → Components & direction and press 3i + 4j + 12k: |a| = 13.
Read the direction cosines and check l² + m² + n² = 1.

Takeaway: a vector is its magnitude times its unit vector, and the direction cosines are the coordinates of that unit vector.

Class 12 — The dot product and projections

Concept: a · b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂. It is a scalar, it is 0 for perpendicular vectors, and the projection of a on b is (a · b) ÷ |b|.

Open Vectors in the plane → Dot product & projection and press perpendicular (a · b = 0).
Press 45° apart and read the angle from cos θ = a · b ÷ (|a||b|).
Drag b around a and watch the projection (teal) shrink to 0 and turn negative.
Open Vectors in space → Dot product & angle and press perpendicular.
Check the sign: a · b positive means an acute angle.

Takeaway: the dot product measures how much two vectors point the same way.

Class 12 — The cross product and the scalar triple product

Concept: a × b is perpendicular to both a and b, with length |a||b| sin θ, which is the area of the parallelogram on a and b. The scalar triple product [a b c] = a · (b × c) is the volume of the parallelepiped on a, b, c, and is 0 when the three vectors are coplanar.

Open Vectors in space → Cross product and press i × j = k.
Turn the scene to check that a × b is perpendicular to the parallelogram.
Read the parallelogram area and the triangle area (half of it).
Tick add a third vector c and press unit cube: the volume is 27 (for 3 × 3 × 3).
Press parallel (a × b = 0) and see the area collapse to 0.

Takeaway: the cross product gives a perpendicular vector and an area; the triple product gives a volume.

Class 12 — The section formula for vectors

Concept: the point P dividing AB internally in the ratio m : n has position vector (m·b + n·a) ÷ (m + n); externally it is (m·b − n·a) ÷ (m − n). The midpoint is the case m = n.

Open Vectors in the plane → Section formula and drag A and B.
Set m = n and check that P is the midpoint.
Set m : n = 3 : 1 and see P close to B.
Tick divide externally and see P outside AB.
Read AP : PB in the cards: it equals m ÷ n.

Takeaway: P = (mB + nA) ÷ (m + n): each end is weighted by the opposite part of the ratio.

Class 12 — Lines in space: equations, angle and shortest distance

Concept: a line is r = a + λb (a point and a direction), or in Cartesian form (x − x₁) ÷ a = (y − y₁) ÷ b = (z − z₁) ÷ c. Two lines in space either meet, are parallel, or are skew. The angle between them is the angle between their directions, and the shortest distance between skew lines is |(a₂ − a₁) · (b₁ × b₂)| ÷ |b₁ × b₂|.

Open Lines & planes → Two lines and press skew lines: the shortest distance is 1 (the red dashed segment).
Press intersecting at (1, 2, 3) and read the point of intersection.
Press parallel lines and then the same line.
Edit line 2 with the sliders until the distance becomes 0: the lines now meet.
Press perpendicular and meeting and read the 90° angle.

Takeaway: lines meet when the shortest distance is 0; skew lines have a positive distance and non-parallel directions.

Class 12 — Planes: equation and distance from a point

Concept: the plane through a point with normal n is n · r = d, or ax + by + cz = d. The distance from a point Q to the plane is |n · Q − d| ÷ |n|, and the foot of the perpendicular lies along the normal.

Open Point & plane and press x + y + z = 3 and Q(3, 2, 1).
Read the distance |6 − 3| ÷ √3 and the foot F of the perpendicular.
Press a point on the plane: the distance is 0.
Change the normal and the value of d and see the plane tilt and move.
Read the intercepts on the axes.

Takeaway: distance = |n · Q − d| ÷ |n|, along the normal.

Class 12 — Angles between a line and a plane, and between two planes

Concept: the angle between a line and a plane is 90° minus the angle between the line and the normal: sin θ = |n · b| ÷ (|n||b|). The angle between two planes is the angle between their normals: cos θ = |n₁ · n₂| ÷ (|n₁||n₂|). Two planes meet in a line with direction n₁ × n₂.

Open Line & plane and press line meets the plane: read the angle and the point where it meets.
Press line parallel to the plane and line inside the plane: n · b = 0 in both.
Open Two planes and press planes at 45°.
Press perpendicular planes and parallel planes.
Read the line of intersection and check that its direction is perpendicular to both normals.

Takeaway: line and plane use the sine of the angle with the normal's complement; plane and plane use the normals' cosine.

Class 12 — The vector triple product and Lagrange's identity

Concept: a × (b × c) is a vector that lies in the plane of b and c, and a × (b × c) = b(a · c) − c(a · b). The scalar triple product [a b c] = a · (b × c) is the volume of the box on a, b, c.

Open Vectors in space → Triple products and press the first preset: a = (1, 2, 3), b = (2, −1, 1), c = (0, 1, 2). Then b × c = (−3, −4, 2) and a × (b × c) = (16, −11, 2).
Check with the rule: a · c = 8 and a · b = 3, so b(a · c) − c(a · b) = 8(2, −1, 1) − 3(0, 1, 2) = (16, −11, 2). The two vectors are equal (the lab shows ✓).
Not associative: (a × b) × c = (15, −10, 5) is a different vector. Its rule is b(a · c) − a(b · c) = 8(2, −1, 1) − 1(1, 2, 3) = (15, −10, 5). The first lies in the plane of b and c, the second in the plane of a and b.
Scalar triple product: [a b c] = −5 here. It does not change when you rotate the order (a, b, c → b, c, a → c, a, b) and changes sign when two vectors swap. The tetrahedron on a, b, c has volume 5/6. Press the third preset (a = (2, 1, 3), b = (1, 1, 1), c = (3, 2, 4)): [a b c] = 0, so the vectors are coplanar.
Lagrange's identity: |a × b|² = |a|²|b|² − (a · b)². For a = (1, 2, 3), b = (2, −1, 1): a × b = (5, 5, −5), 75 = 14 × 6 − 3² = 84 − 9.

Takeaway: “BAC − CAB”: a × (b × c) = b(a · c) − c(a · b). Coplanar vectors have [a b c] = 0.

Class 12 — Point and line, plane through three points, family of planes, common perpendicular

Concept: the standard 3D questions each reduce to a cross product or a dot product. Open Lines & planes.

Distance of a point from a line (view Point & line): the line r = (−2, −1, 0) + λ(1, 2, 2) and Q(2, 3, 1). Then (Q − a) × b = (4, 4, 1) × (1, 2, 2) = (6, −7, 4), whose length is √101 = 10.05, and |b| = 3, so the distance is 10.05 ÷ 3 = 3.35. The foot of the perpendicular is at λ = (Q − a) · b ÷ |b|² = 14/9, F = (−0.44, 2.11, 3.11), and the image of Q is 2F − Q = (−2.89, 1.22, 5.22).
Plane through three points (view Three points): A(1, 0, 0), B(0, 2, 0), C(0, 0, 3). n = AB × AC = (−1, 2, 0) × (−1, 0, 3) = (6, 3, 2), so the plane is 6x + 3y + 2z = 6, in intercept form x/1 + y/2 + z/3 = 1. The area of the triangle is ½|n| = 3.5. For D(1, 1, 1) the distance is 5/7 = 0.714, [AB AC AD] = 5 and the tetrahedron has volume 5/6. Press the second preset to put D on the plane: the distance is 0 and the four points are coplanar.
Family of planes (view Family of planes): P₁: x + y + z = 3 and P₂: x − y = 1. Every plane through their line of intersection is (x + y + z − 3) + λ(x − y − 1) = 0. Through the point (2, 0, 0): −1 + λ(1) = 0, λ = 1, the plane 2x + z = 4. Through the origin: −3 + λ(−1) = 0, λ = −3, the plane −2x + 4y + z = 0. You do not need to find the line first.
Shortest distance between skew lines (view Two lines, the opening example): r = (−3, −2, −1) + λ(2, 1, 1) and r = (1, −3, 2) + μ(0, 1, 1). b₁ × b₂ = (0, −2, 2), (a₂ − a₁) · (b₁ × b₂) = (4, −1, 3) · (0, −2, 2) = 8, so d = 8 ÷ √8 = 2.828. The lab now also gives the feet of the common perpendicular, (1, 0, 1) and (1, −2, 3), whose direction is b₁ × b₂.
Image of a point in a plane (view Point & plane): the foot of Q(3, 2, 1) on x + y + z = 3 is (2, 1, 0), so the image is 2F − Q = (1, 0, −1).

Takeaway: distance from a point to a line = |(Q − a) × b| ÷ |b|; plane through three points has normal AB × AC; family of planes P₁ + λP₂ = 0; skew-line distance = |(a₂ − a₁) · (b₁ × b₂)| ÷ |b₁ × b₂|.

The maths behind it

What each mode covers

Vectors in the plane — two vectors you can drag: the triangle and parallelogram laws for sum and difference, scalar multiples and the unit vector, the dot product with the angle and the projection, and the section formula (internal and external). Class 12.

Vectors in space — a 3D scene you can turn: components, magnitude and direction cosines; sum, difference and scalar multiple; dot product and angle; cross product with the parallelogram area; the scalar triple product and the volume of a box; and the distance and section formulas for points, with the eight octants. Classes 11 and 12.

Lines & planes — two lines (meeting, parallel, coincident or skew, the angle and the shortest distance), a point and a plane (equation, distance, foot of the perpendicular), a line and a plane (angle and meeting point) and two planes (angle and line of intersection). Class 12.

Part of Shunya. See the syllabus map for the chapters these modes map to. In the 3D scenes the z-axis points up, and you can turn the scene by dragging it, with the sliders or with the arrow keys. The two-dimensional coordinate ideas are in the Function & Graph Explorer.

Formulas used in this lab

Points in three dimensions Class 11

Distance between A and B (Δx = x₂ − x₁, and so on): $AB = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$
Distance from the origin: $OP = \sqrt{x^2 + y^2 + z^2}$
Midpoint (the x-coordinate; y and z are the same): $x = \dfrac{x_1 + x_2}{2}$
Internal section m : n (the x-coordinate): $x = \dfrac{mx_2 + nx_1}{m + n}$
External section m : n: $\dfrac{m\,\vec{b} - n\,\vec{a}}{m - n}$

Vectors Class 12

Magnitude: $|\vec{a}| = \sqrt{x^2 + y^2 + z^2}$
Unit vector: $\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}$
Direction cosines: $l = \dfrac{x}{|\vec{a}|},\quad m = \dfrac{y}{|\vec{a}|},\quad n = \dfrac{z}{|\vec{a}|}$
The identity: $l^2 + m^2 + n^2 = 1$
Vector from A to B: $\overrightarrow{AB} = \vec{b} - \vec{a}$
Triangle inequality: $|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|$

Dot product Class 12

Definition: $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$
By components: $\vec{a}\cdot\vec{b} = x_1x_2 + y_1y_2 + z_1z_2$
Angle between vectors: $\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$
Projection of a on b: $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$
Perpendicular: $\vec{a}\cdot\vec{b} = 0$

Cross product Class 12

Magnitude: $|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$
By components: $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$
Area of a parallelogram: $|\vec{a}\times\vec{b}|$
Area of a triangle: $\tfrac{1}{2}\,|\vec{a}\times\vec{b}|$
Order matters: $\vec{a}\times\vec{b} = -\,\vec{b}\times\vec{a}$
Parallel vectors: $\vec{a}\times\vec{b} = \vec{0}$

Scalar triple product Class 12

Definition: $[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c})$
Cyclic: $[\vec{a}\ \vec{b}\ \vec{c}] = [\vec{b}\ \vec{c}\ \vec{a}]$
Volume of a parallelepiped: $V = |[\vec{a}\ \vec{b}\ \vec{c}]|$
Volume of a tetrahedron: $V = \tfrac{1}{6}\,|[\vec{a}\ \vec{b}\ \vec{c}]|$
Coplanar vectors: $[\vec{a}\ \vec{b}\ \vec{c}] = 0$

Lines in space Class 12

Vector form: $\vec{r} = \vec{a} + \lambda\vec{b}$
Cartesian form: $\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}$
Through two points: $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$
Angle between two lines: $\cos\theta = \dfrac{|\vec{b}_1\cdot\vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}$
Perpendicular and parallel: $\vec{b}_1\cdot\vec{b}_2 = 0,\qquad \vec{b}_1\times\vec{b}_2 = \vec{0}$

Shortest distance between lines Class 12

Skew lines: $d = \dfrac{|(\vec{a}_2 - \vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)|}{|\vec{b}_1\times\vec{b}_2|}$
Parallel lines: $d = \dfrac{|(\vec{a}_2 - \vec{a}_1)\times\vec{b}|}{|\vec{b}|}$
Lines that meet: $(\vec{a}_2 - \vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2) = 0$
Parallel directions: $\vec{b}_1\times\vec{b}_2 = \vec{0}$

Planes Class 12

Vector form: $\vec{r}\cdot\vec{n} = d$
Cartesian form: $ax + by + cz = d$
Through three points: $\vec{n} = (\vec{b} - \vec{a})\times(\vec{c} - \vec{a})$
Intercept form: $\dfrac{x}{p} + \dfrac{y}{q} + \dfrac{z}{r} = 1$
Distance of a point from a plane: $\dfrac{|\vec{n}\cdot\vec{Q} - d|}{|\vec{n}|}$

Angles with planes Class 12

Line and plane: $\sin\theta = \dfrac{|\vec{n}\cdot\vec{b}|}{|\vec{n}||\vec{b}|}$
Two planes: $\cos\theta = \dfrac{|\vec{n}_1\cdot\vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$
Line of intersection: $\text{direction } \vec{n}_1\times\vec{n}_2$
Perpendicular planes: $\vec{n}_1\cdot\vec{n}_2 = 0$

Triple products and the standard 3D distances Class 12

Vector triple product: $\vec a \times (\vec b \times \vec c) = \vec b(\vec a\cdot\vec c) - \vec c(\vec a\cdot\vec b)$
Lagrange's identity: $|\vec a\times\vec b|^2 = |\vec a|^2|\vec b|^2 - (\vec a\cdot\vec b)^2$
Scalar triple product: $[\vec a\ \vec b\ \vec c] = \vec a\cdot(\vec b\times\vec c) = [\vec b\ \vec c\ \vec a] = -[\vec b\ \vec a\ \vec c]$
Coplanar vectors and tetrahedron volume: $[\vec a\ \vec b\ \vec c] = 0;\qquad V_{\text{tetra}} = \tfrac16\,|[\overrightarrow{AB}\ \overrightarrow{AC}\ \overrightarrow{AD}]|$
Distance of a point from a line: $d = \dfrac{|(\vec q - \vec a)\times\vec b|}{|\vec b|}$
Plane through three points: $\vec n = \overrightarrow{AB}\times\overrightarrow{AC},\quad \vec n\cdot(\vec r - \vec A) = 0$
Family of planes through a line: $P_1 + \lambda P_2 = 0$
Shortest distance between skew lines: $d = \dfrac{|(\vec a_2 - \vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}$