H2 Maths Notes (JC 1-2): 3) Vectors

Before you begin
Consolidate IP vector algebra (column vectors, magnitude, simple dot product) so JC notation feels familiar. Keep your GC in Exam Mode and know where to find the vector, matrix, and simultaneous equation menus.

3.1 | Vector Language

Core ideas

• A vector has magnitude and direction; write it as a column ( \begin{pmatrix} a \\ b \\ c \end{pmatrix} ) or in component form ( a \space \mathbf{i} + b\space\mathbf{j} + c\space\mathbf{k} ).
• The magnitude of vector ( \vec{a} ) with components ( (a_1, a_2, a_3) ) is ( |\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} ).
• Two vectors are parallel when one is a scalar multiple of the other.
• A line in 3-D can be written as ( \vec{r} = \vec{a} + \lambda \vec{d} ) where ( \vec{a} ) is a position vector on the line and ( \vec{d} ) its direction vector.
• A plane can be expressed as ( \vec{r} \cdot \vec{n} = k ) or in Cartesian \( ax + by + cz = d \) with normal ( \vec{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} ).

Example 1 -- Checking parallel vectors

Given ( \vec{p} = \begin{pmatrix} 3 \\ -6 \\ 9 \end{pmatrix} ) and ( \vec{q} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} ), show they are parallel.

• Observe ( \vec{p} = 3 \vec{q} ).
• Conclude ( \vec{p} ) and ( \vec{q} ) have the same direction.

3.2 | Scalar and Vector Products

Scalar (dot) product

• Definition: ( \vec{a} \cdot \vec{b} = |\vec{a}|\space|\vec{b}| \cos \theta ).
• Algebraic form: multiply corresponding components and add.
• Uses: angle between vectors, projection, perpendicularity test (dot product zero).

Vector (cross) product

• Definition: ( \vec{a} \times \vec{b} ) is a vector perpendicular to both ( \vec{a} ) and ( \vec{b} ) with magnitude ( |\vec{a}|\space|\vec{b}| \sin \theta ).
• Determinant shortcut: [ \vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}. ]
• Uses: find normals, areas of parallelograms ( |\vec{a} \times \vec{b}| ).

Mixed product and volume

• Triple scalar product ( (\vec{a} \times \vec{b}) \cdot \vec{c} ) gives the volume of the parallelepiped formed by the three vectors.
• If the triple product is zero, the vectors are coplanar.

3.3 | Geometry and Modelling

Line and plane relationships

• Intersection: solve simultaneously to find common points.
• Parallel: direction vector parallel to plane normal implies no intersection unless the point lies on the plane.
• Perpendicular: line direction dot product with plane normal equals zero.

Shortest distances

• Point to line: use projection or the formula based on cross product ( \frac{|\vec{AP} \times \vec{d}|}{|\vec{d}|} ).
• Point to plane: distance is ( \frac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2 + b^2 + c^2}} ).
• Line to line: if skew, use ( (\vec{d_1} \times \vec{d_2}) ) to build a normal and project the connecting vector.

Example 2 -- Line-plane intersection

Find the intersection of line ( \vec{r} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix} ) with plane \( 2x - y + z = 9 \).

1. Substitute parametric coordinates into the plane: ( 2(2 + \lambda) - (1 - 2\lambda) + (-3 + 4\lambda) = 9 ).
2. Simplify to solve for ( \lambda = 2 ).
3. Substitute back to get intersection point ( (4, -3, 5) ).

Example 3 -- Shortest distance from point to plane

Point ( P(3, -1, 2) ) and plane \( 3x + 2y - z = 7 \).

1. Compute numerator ( |3(3) + 2(-1) - 1(2) - 7| = |9 - 2 - 2 - 7| = 2 ).
2. Denominator ( \sqrt{3^2 + 2^2 + (-1)^2} = \sqrt{14} ).
3. Distance ( \frac{2}{\sqrt{14}} ) units.

3.4 | Calculator Workflows

• Store direction vectors as lists or matrices to speed up dot and cross products.
• Use simultaneous equation solvers for intersection problems after substituting parameters.
• For distance checks, program short GC functions (e.g. DISTPLANE(x, y, z, a, b, c, d)) so you can evaluate quickly while still showing manual setup in working.

3.5 | Exam Watch Points

• Always state whether vectors are position vectors (from origin) or free vectors.
• Show the step where you substitute line coordinates into plane equations; markers look for the substitution before solving.
• When quoting magnitudes, keep exact forms (( \sqrt{14} )) until the final step unless the question asks for decimals.
• Interpret your answers: mention direction for shortest distance vectors and highlight parallel or perpendicular conclusions explicitly.

3.6 | Quick Revision Checklist

• [ ] Switch comfortably between column, component, and vector equation forms.
• [ ] Compute dot, cross, and triple products accurately under time pressure.
• [ ] Solve line-line, line-plane, and plane-plane systems with clear parameter steps.
• [ ] Explain geometric meaning (parallel, perpendicular, skew) in complete sentences.

Next step: expand Topic 3 into sub-topics 3.1 to 3.3 with exam-grade examples and modelling tasks.